Resolution of Surfaces

Introduction

This chapter discusses resolution of singularities of surfaces following Lipman [Lipman] and mostly following the exposition of Artin in [Artin-Lipman]. The main result (Theorem 0BGP) tells us that a Noetherian \(2\)-dimensional scheme \(Y\) has a resolution of singularities when it has a finite normalization \(Y^\nu \to Y\) with finitely many singular points \(y_i \in Y^\nu\) and for each \(i\) the completion \(\mathcal{O}_{Y^\nu, y_i}^\wedge\) is normal.

To be sure, if \(Y\) is a \(2\)-dimensional scheme of finite type over a quasi-excellent base ring \(R\) (for example a field or a Dedekind domain with fraction field of characteristic \(0\) such as \(\mathbf{Z}\)) then the normalization of \(Y\) is finite, has finitely many singular points, and the completions of the local rings are normal. See the discussion in More on Algebra, Sections 07P6, 07GG, and 07QS and More on Algebra, Lemma 0BFK. Thus such a \(Y\) has a resolution of singularities.

A rough outline of the proof is as follows. Let \(A\) be a Noetherian local domain of dimension \(2\). The steps of the proof are as follows

1. replace \(A\) by its normalization,

2. prove Grauert-Riemenschneider,

3. show there is a maximum \(g\) of the lengths of \(H^1(X, \mathcal{O}_X)\) over all normal modifications \(X \to \Spec(A)\) and reduce to the case \(g = 0\),

4. we say \(A\) defines a rational singularity if \(g = 0\) and in this case after a finite number of blowups we may assume \(A\) is Gorenstein and \(g = 0\),

5. we say \(A\) defines a rational double point if \(g = 0\) and \(A\) is Gorenstein and in this case we explicitly resolve singularities.

Each of these steps needs assumptions on the ring \(A\). We will discuss each of these in turn.

Ad N: Here we need to assume that \(A\) has a finite normalization (this is not automatic). Throughout most of the chapter we will assume that our scheme is Nagata if we need to know some normalization is finite. However, being Nagata is a slightly stronger condition than is given to us in the statement of the theorem. A solution to this (slight) problem would have been to use that our ring \(A\) is formally unramified (i.e., its completion is reduced) and to use Lemma 0BG8. However, the way our proof works, it turns out it is easier to use Lemma 0BG9 to lift finiteness of the normalization over the completion to finiteness of the normalization over \(A\).

Ad V: This is Proposition 0AXD and it roughly states that for a normal modification \(f : X \to \Spec(A)\) one has \(R^1f_*\omega_X = 0\) where \(\omega_X\) is the dualizing module of \(X/A\) (Remark 0B4R). In fact, by duality the result is equivalent to a statement (Lemma 0AXC) about the object \(Rf_*\mathcal{O}_X\) in the derived category \(D(A)\). Having said this, the proof uses the standard fact that components of the special fibre have positive conormal sheaves (Lemma 0AXA).

Ad B: This is in some sense the most subtle part of the proof. In the end we only need to use the output of this step when \(A\) is a complete Noetherian local ring, although the writeup is a bit more general. The terminology is set in Definition 0B4N. If \(g\) (as defined above) is bounded, then a straightforward argument shows that we can find a normal modification \(X \to \Spec(A)\) such that all singular points of \(X\) are rational singularities, see Lemma 0B4P. We show that given a finite extension \(A \subset B\), then \(g\) is bounded for \(B\) if it is bounded for \(A\) in the following two cases: (1) if the fraction field extension is separable, see Lemma 0B4P and (2) if the fraction field extension has degree \(p\), the characteristic is \(p\), and \(A\) is regular and complete, see Lemma 0B4U.

Ad R: Here we reduce the case \(g = 0\) to the Gorenstein case. A marvellous fact, which makes everything work, is that the blowing up of a rational surface singularity is normal, see Lemma 0B4Z.

Ad D: The resolution of rational double points proceeds more or less by hand, see Section 0BGB. A rational double point is a hypersurface singularity (this is true but we don’t prove it as we don’t need it). The local equation looks like \[a_{11} x_1^2 + a_{12} x_1x_2 + a_{13}x_1x_3 + a_{22} x_2^2 + a_{23} x_2x_3 + a_{33} x_3^2 = \sum a_{ijk} x_ix_jx_k\] Using that the quadratic part cannot be zero because the multiplicity is \(2\) and remains \(2\) after any blowup and the fact that every blowup is normal one quickly achieves a resolution. One twist is that we do not have an invariant which decreases every blowup, but we rely on the material on formal arcs from Section 0BG1 to demonstrate that the process stops.

To put everything together some additional work has to be done. The main kink is that we want to lift a resolution of the completion \(A^\wedge\) to a resolution of \(\Spec(A)\). In order to do this we first show that if a resolution exists, then there is a resolution by normalized blowups (Lemma 0BGM). A sequence of normalized blowups can be lifted from the completion by Lemma 0BGA. We then use this even in the proof of resolution of complete local rings \(A\) because our strategy works by induction on the degree of a finite inclusion \(A_0 \subset A\) with \(A_0\) regular, see Lemma 0BGN. With a stronger result in B (such as is proved in Lipman’s paper) this step could be avoided.

A trace map in positive characteristic

Some of the results in this section can be deduced from the much more general discussion on traces on differential forms in de Rham Cohomology, Section 0FK6. See Remark 0FLF for a discussion.

We fix a prime number \(p\). Let \(R\) be an \(\mathbf{F}_p\)-algebra. Given an \(a \in R\) set \(S = R[x]/(x^p - a)\). Define an \(R\)-linear map \[\text{Tr}_x : \Omega_{S/R} \longrightarrow \Omega_R\] by the rule \[x^i\text{d}x \longmapsto \left\{ \begin{matrix} 0 & \text{if} & 0 \leq i \leq p - 2, \\ \text{d}a & \text{if} & i = p - 1 \end{matrix} \right.\] This makes sense as \(\Omega_{S/R}\) is a free \(R\)-module with basis \(x^i\text{d}x\), \(0 \leq i \leq p - 1\). The following lemma implies that the trace map is well defined, i.e., independent of the choice of the coordinate \(x\).

Lemma

Let \(\varphi : R[x]/(x^p - a) \to R[y]/(y^p - b)\) be an \(R\)-algebra homomorphism. Then \(\text{Tr}_x = \text{Tr}_y \circ \varphi\).

Proof

Say \(\varphi(x) = \lambda_0 + \lambda_1 y + \ldots + \lambda_{p - 1}y^{p - 1}\) with \(\lambda_i \in R\). The condition that mapping \(x\) to \(\lambda_0 + \lambda_1 y + \ldots + \lambda_{p - 1}y^{p - 1}\) induces an \(R\)-algebra homomorphism \(R[x]/(x^p - a) \to R[y]/(y^p - b)\) is equivalent to the condition that \[a = \lambda_0^p + \lambda_1^p b + \ldots + \lambda_{p - 1}^pb^{p - 1}\] in the ring \(R\). Consider the polynomial ring \[R_{univ} = \mathbf{F}_p[b, \lambda_0, \ldots, \lambda_{p - 1}]\] with the element \(a = \lambda_0^p + \lambda_1^p b + \ldots + \lambda_{p - 1}^pb^{p - 1}\) Consider the universal algebra map \(\varphi_{univ} : R_{univ}[x]/(x^p - a) \to R_{univ}[y]/(y^p - b)\) given by mapping \(x\) to \(\lambda_0 + \lambda_1 y + \ldots + \lambda_{p - 1}y^{p - 1}\). We obtain a canonical map \[R_{univ} \longrightarrow R\] sending \(b, \lambda_i\) to \(b, \lambda_i\). By construction we get a commutative diagram \[\xymatrix{ R_{univ}[x]/(x^p - a) \ar[r] \ar[d]_{\varphi_{univ}} & R[x]/(x^p - a) \ar[d]^\varphi \\ R_{univ}[y]/(y^p - b) \ar[r] & R[y]/(y^p - b) }\] and the horizontal arrows are compatible with the trace maps. Hence it suffices to prove the lemma for the map \(\varphi_{univ}\). Thus we may assume \(R = \mathbf{F}_p[b, \lambda_0, \ldots, \lambda_{p - 1}]\) is a polynomial ring. We will check the lemma holds in this case by evaluating \(\text{Tr}_y(\varphi(x)^i\text{d}\varphi(x))\) for \(i = 0 , \ldots, p - 1\).

The case \(0 \leq i \leq p - 2\). Expand \[(\lambda_0 + \lambda_1 y + \ldots + \lambda_{p - 1}y^{p - 1})^i (\lambda_1 + 2 \lambda_2 y + \ldots + (p - 1)\lambda_{p - 1}y^{p - 2})\] in the ring \(R[y]/(y^p - b)\). We have to show that the coefficient of \(y^{p - 1}\) is zero. For this it suffices to show that the expression above as a polynomial in \(y\) has vanishing coefficients in front of the powers \(y^{pk - 1}\). Then we write our polynomial as \[\frac{\text{d}}{(i + 1)\text{d}y} (\lambda_0 + \lambda_1 y + \ldots + \lambda_{p - 1}y^{p - 1})^{i + 1}\] and indeed the coefficients of \(y^{kp - 1}\) are all zero.

The case \(i = p - 1\). Expand \[(\lambda_0 + \lambda_1 y + \ldots + \lambda_{p - 1}y^{p - 1})^{p - 1} (\lambda_1 + 2 \lambda_2 y + \ldots + (p - 1)\lambda_{p - 1}y^{p - 2})\] in the ring \(R[y]/(y^p - b)\). To finish the proof we have to show that the coefficient of \(y^{p - 1}\) times \(\text{d}b\) is \(\text{d}a\). Here we use that \(R\) is \(S/pS\) where \(S = \mathbf{Z}[b, \lambda_0, \ldots, \lambda_{p - 1}]\). Then the above, as a polynomial in \(y\), is equal to \[\frac{\text{d}}{p\text{d}y} (\lambda_0 + \lambda_1 y + \ldots + \lambda_{p - 1}y^{p - 1})^p\] Since \(\frac{\text{d}}{\text{d}y}(y^{pk}) = pk y^{pk - 1}\) it suffices to understand the coefficients of \(y^{pk}\) in the polynomial \((\lambda_0 + \lambda_1 y + \ldots + \lambda_{p - 1}y^{p - 1})^p\) modulo \(p\). The sum of these terms gives \[\lambda_0^p + \lambda_1^py^p + \ldots + \lambda_{p - 1}^py^{p(p - 1)} \bmod p\] Whence we see that we obtain after applying the operator \(\frac{\text{d}}{p\text{d}y}\) and after reducing modulo \(y^p - b\) the value \[\lambda_1^p + 2\lambda_2^pb + \ldots + (p - 1)\lambda_{p - 1}b^{p - 2}\] for the coefficient of \(y^{p - 1}\) we wanted to compute. Now because \(a = \lambda_0^p + \lambda_1^p b + \ldots + \lambda_{p - 1}^pb^{p - 1}\) in \(R\) we obtain that \[\text{d}a = (\lambda_1^p + 2 \lambda_2^p b + \ldots + (p - 1) \lambda_{p - 1}^p b^{p - 2}) \text{d}b\] in \(R\). This proves that the coefficient of \(y^{p - 1}\) is as desired.

Lemma

Let \(\mathbf{F}_p \subset \Lambda \subset R \subset S\) be ring extensions and assume that \(S\) is isomorphic to \(R[x]/(x^p - a)\) for some \(a \in R\). Then there are canonical \(R\)-linear maps \[\text{Tr} : \Omega^{t + 1}_{S/\Lambda} \longrightarrow \Omega_{R/\Lambda}^{t + 1}\] for \(t \geq 0\) such that \[\eta_1 \wedge \ldots \wedge \eta_t \wedge x^i\text{d}x \longmapsto \left\{ \begin{matrix} 0 & \text{if} & 0 \leq i \leq p - 2, \\ \eta_1 \wedge \ldots \wedge \eta_t \wedge \text{d}a & \text{if} & i = p - 1 \end{matrix} \right.\] for \(\eta_i \in \Omega_{R/\Lambda}\) and such that \(\text{Tr}\) annihilates the image of \(S \otimes_R \Omega_{R/\Lambda}^{t + 1} \to \Omega_{S/\Lambda}^{t + 1}\).

Proof

For \(t = 0\) we use the composition \[\Omega_{S/\Lambda} \to \Omega_{S/R} \to \Omega_R \to \Omega_{R/\Lambda}\] where the second map is Lemma 0ADZ. There is an exact sequence \[H_1(L_{S/R}) \xrightarrow{\delta} \Omega_{R/\Lambda} \otimes_R S \to \Omega_{S/\Lambda} \to \Omega_{S/R} \to 0\] (Algebra, Lemma 00S2). The module \(\Omega_{S/R}\) is free over \(S\) with basis \(\text{d}x\) and the module \(H_1(L_{S/R})\) is free over \(S\) with basis \(x^p - a\) which \(\delta\) maps to \(-\text{d}a \otimes 1\) in \(\Omega_{R/\Lambda} \otimes_R S\). In particular, if we set \[M = \Coker(R \to \Omega_{R/\Lambda}, 1 \mapsto -\text{d}a)\] then we see that \(\Coker(\delta) = M \otimes_R S\). We obtain a canonical map \[\Omega^{t + 1}_{S/\Lambda} \to \wedge_S^t(\Coker(\delta)) \otimes_S \Omega_{S/R} = \wedge^t_R(M) \otimes_R \Omega_{S/R}\] Now, since the image of the map \(\text{Tr} : \Omega_{S/R} \to \Omega_{R/\Lambda}\) of Lemma 0ADZ is contained in \(R\text{d}a\) we see that wedging with an element in the image annihilates \(\text{d}a\). Hence there is a canonical map \[\wedge^t_R(M) \otimes_R \Omega_{S/R} \to \Omega_{R/\Lambda}^{t + 1}\] mapping \(\overline{\eta}_1 \wedge \ldots \wedge \overline{\eta}_t \wedge \omega\) to \(\eta_1 \wedge \ldots \wedge \eta_t \wedge \text{Tr}(\omega)\).

Remark

Let \(\mathbf{F}_p \subset \Lambda \subset R \subset S\) and \(\text{Tr}\) be as in Lemma 0AX5. By de Rham Cohomology, Proposition 0FLC there is a canonical map of complexes \[\Theta_{S/R} : \Omega_{S/\Lambda}^\bullet \longrightarrow \Omega_{R/\Lambda}^\bullet\] The computation in de Rham Cohomology, Example 0FLD shows that \(\Theta_{S/R}(x^i \text{d}x) = \text{Tr}_x(x^i\text{d}x)\) for all \(i\). Since \(\text{Trace}_{S/R} = \Theta^0_{S/R}\) is identically zero and since \[\Theta_{S/R}(a \wedge b) = a \wedge \Theta_{S/R}(b)\] for \(a \in \Omega^i_{R/\Lambda}\) and \(b \in \Omega^j_{S/\Lambda}\) it follows that \(\text{Tr} = \Theta_{S/R}\). The advantage of using \(\text{Tr}\) is that it is a good deal more elementary to construct.

Lemma

Let \(S\) be a scheme over \(\mathbf{F}_p\). Let \(f : Y \to X\) be a finite morphism of Noetherian normal integral schemes over \(S\). Assume

1. the extension of function fields is purely inseparable of degree \(p\), and

2. \(\Omega_{X/S}\) is a coherent \(\mathcal{O}_X\)-module (for example if \(X\) is of finite type over \(S\)).

For \(i \geq 1\) there is a canonical map \[\text{Tr} : f_*\Omega^i_{Y/S} \longrightarrow (\Omega_{X/S}^i)^{**}\] whose stalk in the generic point of \(X\) recovers the trace map of Lemma 0AX5.

Proof

The exact sequence \(f^*\Omega_{X/S} \to \Omega_{Y/S} \to \Omega_{Y/X} \to 0\) shows that \(\Omega_{Y/S}\) and hence \(f_*\Omega_{Y/S}\) are coherent modules as well. Thus it suffices to prove the trace map in the generic point extends to stalks at \(x \in X\) with \(\dim(\mathcal{O}_{X, x}) = 1\), see Divisors, Lemma 0AY7. Thus we reduce to the case discussed in the next paragraph.

Assume \(X = \Spec(A)\) and \(Y = \Spec(B)\) with \(A\) a discrete valuation ring and \(B\) finite over \(A\). Since the induced extension \(L/K\) of fraction fields is purely inseparable, we see that \(B\) is local too. Hence \(B\) is a discrete valuation ring too. Then either

1. \(B/A\) has ramification index \(p\) and hence \(B = A[x]/(x^p - a)\) where \(a \in A\) is a uniformizer, or

2. \(\mathfrak m_B = \mathfrak m_A B\) and the residue field \(B/\mathfrak m_A B\) is purely inseparable of degree \(p\) over \(\kappa_A = A/\mathfrak m_A\). Choose any \(x \in B\) whose residue class is not in \(\kappa_A\) and then we’ll have \(B = A[x]/(x^p - a)\) where \(a \in A\) is a unit.

Let \(\Spec(\Lambda) \subset S\) be an affine open such that \(X\) maps into \(\Spec(\Lambda)\). Then we can apply Lemma 0AX5 to see that the trace map extends to \(\Omega^i_{B/\Lambda} \to \Omega^i_{A/\Lambda}\) for all \(i \geq 1\).

Quadratic transformations

In this section we study what happens when we blow up a nonsingular point on a surface. We hesitate the formally define such a morphism as a quadratic transformation as on the one hand often other names are used and on the other hand the phrase “quadratic transformation” is sometimes used with a different meaning.

Lemma

Let \((A, \mathfrak m, \kappa)\) be a regular local ring of dimension \(2\). Let \(f : X \to S = \Spec(A)\) be the blowing up of \(A\) in \(\mathfrak m\) wotj exceptional divisor \(E\). There is a closed immersion \[r : X \longrightarrow \mathbf{P}^1_S\] over \(S\) such that

1. \(r|_E : E \to \mathbf{P}^1_\kappa\) is an isomorphism,

2. \(\mathcal{O}_X(E) = \mathcal{O}_X(-1) = r^*\mathcal{O}_{\mathbf{P}^1}(-1)\), and

3. \(\mathcal{C}_{E/X} = (r|_E)^*\mathcal{O}_{\mathbf{P}^1}(1)\) and \(\mathcal{N}_{E/X} = (r|_E)^*\mathcal{O}_{\mathbf{P}^1}(-1)\).

Proof

As \(A\) is regular of dimension \(2\) we can write \(\mathfrak m = (x, y)\). Then \(x\) and \(y\) placed in degree \(1\) generate the Rees algebra \(\bigoplus_{n \geq 0} \mathfrak m^n\) over \(A\). Recall that \(X = \text{Proj}(\bigoplus_{n \geq 0} \mathfrak m^n)\), see Divisors, Lemma 0804. Thus the surjection \[A[T_0, T_1] \longrightarrow \bigoplus\nolimits_{n \geq 0} \mathfrak m^n, \quad T_0 \mapsto x,\ T_1 \mapsto y\] of graded \(A\)-algebras induces a closed immersion \(r : X \to \mathbf{P}^1_S = \text{Proj}(A[T_0, T_1])\) such that \(\mathcal{O}_X(1) = r^*\mathcal{O}_{\mathbf{P}^1_S}(1)\), see Constructions, Lemma 01N1. This proves (2) because \(\mathcal{O}_X(E) = \mathcal{O}_X(-1)\) by Divisors, Lemma 02OS.

To prove (1) note that \[\left(\bigoplus\nolimits_{n \geq 0} \mathfrak m^n\right) \otimes_A \kappa = \bigoplus\nolimits_{n \geq 0} \mathfrak m^n/\mathfrak m^{n + 1} \cong \kappa[\overline{x}, \overline{y}]\] a polynomial algebra, see Algebra, Lemma 00NO. This proves that the fibre of \(X \to S\) over \(\Spec(\kappa)\) is equal to \(\text{Proj}(\kappa[\overline{x}, \overline{y}]) = \mathbf{P}^1_\kappa\), see Constructions, Lemma 01N2. Recall that \(E\) is the closed subscheme of \(X\) defined by \(\mathfrak m\mathcal{O}_X\), i.e., \(E = X_\kappa\). By our choice of the morphism \(r\) we see that \(r|_E\) in fact produces the identification of \(E = X_\kappa\) with the special fibre of \(\mathbf{P}^1_S \to S\).

Part (3) follows from (1) and (2) and Divisors, Lemma 0B3P.

Lemma

Let \((A, \mathfrak m, \kappa)\) be a regular local ring of dimension \(2\). Let \(f : X \to S = \Spec(A)\) be the blowing up of \(A\) in \(\mathfrak m\). Then \(X\) is an irreducible regular scheme.

Proof

Observe that \(X\) is integral by Divisors, Lemma 02ND and Algebra, Lemma 00NP. To see \(X\) is regular it suffices to check that \(\mathcal{O}_{X, x}\) is regular for closed points \(x \in X\), see Properties, Lemma 02IT. Let \(x \in X\) be a closed point. Since \(f\) is proper \(x\) maps to \(\mathfrak m\), i.e., \(x\) is a point of the exceptional divisor \(E\). Then \(E\) is an effective Cartier divisor and \(E \cong \mathbf{P}^1_\kappa\). Thus if \(g \in \mathfrak m_x \subset \mathcal{O}_{X, x}\) is a local equation for \(E\), then \(\mathcal{O}_{X, x}/(g) \cong \mathcal{O}_{\mathbf{P}^1_\kappa, x}\). Since \(\mathbf{P}^1_\kappa\) is covered by two affine opens which are the spectrum of a polynomial ring over \(\kappa\), we see that \(\mathcal{O}_{\mathbf{P}^1_\kappa, x}\) is regular by Algebra, Lemma 00OP. We conclude by Algebra, Lemma 00NU.

Lemma

Let \((A, \mathfrak m, \kappa)\) be a regular local ring of dimension \(2\). Let \(f : X \to S = \Spec(A)\) be the blowing up of \(A\) in \(\mathfrak m\). Then \(\Pic(X) = \mathbf{Z}\) generated by \(\mathcal{O}_X(E)\).

Proof

Recall that \(E = \mathbf{P}^1_\kappa\) has Picard group \(\mathbf{Z}\) with generator \(\mathcal{O}(1)\), see Divisors, Lemma 0BXJ. By Lemma 0AGQ the invertible \(\mathcal{O}_X\)-module \(\mathcal{O}_X(E)\) restricts to \(\mathcal{O}(-1)\). Hence \(\mathcal{O}_X(E)\) generates an infinite cyclic group in \(\Pic(X)\). Since \(A\) is regular it is a UFD, see More on Algebra, Lemma 0AG0. Then the punctured spectrum \(U = S \setminus \{\mathfrak m\} = X \setminus E\) has trivial Picard group, see Divisors, Lemma 0BDA. Hence for every invertible \(\mathcal{O}_X\)-module \(\mathcal{L}\) there is an isomorphism \(s : \mathcal{O}_U \to \mathcal{L}|_U\). Then \(s\) is a regular meromorphic section of \(\mathcal{L}\) and we see that \(\text{div}_\mathcal{L}(s) = nE\) for some \(n \in \mathbf{Z}\) (Divisors, Definition 0BE6). By Divisors, Lemma 0BE8 (and the fact that \(X\) is normal by Lemma 0AGR) we conclude that \(\mathcal{L} = \mathcal{O}_X(nE)\).

Lemma

Let \((A, \mathfrak m, \kappa)\) be a regular local ring of dimension \(2\). Let \(f : X \to S = \Spec(A)\) be the blowing up of \(A\) in \(\mathfrak m\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module.

1. \(H^p(X, \mathcal{F}) = 0\) for \(p \not \in \{0, 1\}\),

2. \(H^1(X, \mathcal{O}_X(n)) = 0\) for \(n \geq -1\),

3. \(H^1(X, \mathcal{F}) = 0\) if \(\mathcal{F}\) or \(\mathcal{F}(1)\) is globally generated,

4. \(H^0(X, \mathcal{O}_X(n)) = \mathfrak m^{\max(0, n)}\),

5. \(\text{length}_A H^1(X, \mathcal{O}_X(n)) = -n(-n - 1)/2\) if \(n < 0\).

Proof

If \(\mathfrak m = (x, y)\), then \(X\) is covered by the spectra of the affine blowup algebras \(A[\frac{\mathfrak m}{x}]\) and \(A[\frac{\mathfrak m}{y}]\) because \(x\) and \(y\) placed in degree \(1\) generate the Rees algebra \(\bigoplus \mathfrak m^n\) over \(A\). See Divisors, Lemma 0804 and Constructions, Lemma 01MD. Since \(X\) is separated by Constructions, Lemma 01MC we see that cohomology of quasi-coherent sheaves vanishes in degrees \(\geq 2\) by Cohomology of Schemes, Lemma 01XI.

Let \(i : E \to X\) be the exceptional divisor, see Divisors, Definition 01OG. Recall that \(\mathcal{O}_X(-E) = \mathcal{O}_X(1)\) is \(f\)-relatively ample, see Divisors, Lemma 02OS. Hence we know that \(H^1(X, \mathcal{O}_X(-nE)) = 0\) for some \(n > 0\), see Cohomology of Schemes, Lemma 02O1. Consider the filtration \[\mathcal{O}_X(-nE) \subset \mathcal{O}_X(-(n - 1)E) \subset \ldots \subset \mathcal{O}_X(-E) \subset \mathcal{O}_X \subset \mathcal{O}_X(E)\] The successive quotients are the sheaves \[\mathcal{O}_X(-t E)/\mathcal{O}_X(-(t + 1)E) = \mathcal{O}_X(t)/\mathcal{I}(t) = i_*\mathcal{O}_E(t)\] where \(\mathcal{I} = \mathcal{O}_X(-E)\) is the ideal sheaf of \(E\). By Lemma 0AGQ we have \(E = \mathbf{P}^1_\kappa\) and \(\mathcal{O}_E(1)\) indeed corresponds to the usual Serre twist of the structure sheaf on \(\mathbf{P}^1\). Hence the cohomology of \(\mathcal{O}_E(t)\) vanishes in degree \(1\) for \(t \geq -1\), see Cohomology of Schemes, Lemma 01XT. Since this is equal to \(H^1(X, i_*\mathcal{O}_E(t))\) (by Cohomology of Schemes, Lemma 089W) we find that \(H^1(X, \mathcal{O}_X(-(t + 1)E)) \to H^1(X, \mathcal{O}_X(-tE))\) is surjective for \(t \geq -1\). Hence \[0 = H^1(X, \mathcal{O}_X(-nE)) \longrightarrow H^1(X, \mathcal{O}_X(-tE)) = H^1(X, \mathcal{O}_X(t))\] is surjective for \(t \geq -1\) which proves (2).

Let \(\mathcal{F}\) be globally generated. This means there exists a short exact sequence \[0 \to \mathcal{G} \to \bigoplus\nolimits_{i \in I} \mathcal{O}_X \to \mathcal{F} \to 0\] Note that \(H^1(X, \bigoplus_{i \in I} \mathcal{O}_X) = \bigoplus_{i \in I} H^1(X, \mathcal{O}_X)\) by Cohomology, Lemma 01FF. By part (2) we have \(H^1(X, \mathcal{O}_X) = 0\). If \(\mathcal{F}(1)\) is globally generated, then we can find a surjection \(\bigoplus_{i \in I} \mathcal{O}_X(-1) \to \mathcal{F}\) and argue in a similar fashion. In other words, part (3) follows from part (2).

For part (4) we note that for all \(n\) large enough we have \(\Gamma(X, \mathcal{O}_X(n)) = \mathfrak m^n\), see Cohomology of Schemes, Lemma 0AG7. If \(n \geq 0\), then we can use the short exact sequence \[0 \to \mathcal{O}_X(n) \to \mathcal{O}_X(n - 1) \to i_*\mathcal{O}_E(n - 1) \to 0\] and the vanishing of \(H^1\) for the sheaf on the left to get a commutative diagram \[\xymatrix{ 0 \ar[r] & \mathfrak m^{\max(0, n)} \ar[r] \ar[d] & \mathfrak m^{\max(0, n - 1)} \ar[r] \ar[d] & \mathfrak m^{\max(0, n)}/\mathfrak m^{\max(0, n - 1)} \ar[r] \ar[d] & 0\\ 0 \ar[r] & \Gamma(X, \mathcal{O}_X(n)) \ar[r] & \Gamma(X, \mathcal{O}_X(n - 1)) \ar[r] & \Gamma(E, \mathcal{O}_E(n - 1)) \ar[r] & 0 }\] with exact rows. In fact, the rows are exact also for \(n < 0\) because in this case the groups on the right are zero. In the proof of Lemma 0AGQ we have seen that the right vertical arrow is an isomorphism (details omitted). Hence if the left vertical arrow is an isomorphism, so is the middle one. In this way we see that (4) holds by descending induction on \(n\).

Finally, we prove (5) by descending induction on \(n\) and the sequences \[0 \to \mathcal{O}_X(n) \to \mathcal{O}_X(n - 1) \to i_*\mathcal{O}_E(n - 1) \to 0\] Namely, for \(n \geq -1\) we already know \(H^1(X, \mathcal{O}_X(n)) = 0\). Since \[H^1(X, i_*\mathcal{O}_E(-2)) = H^1(E, \mathcal{O}_E(-2)) = H^1(\mathbf{P}^1_\kappa, \mathcal{O}(-2)) \cong \kappa\] by Cohomology of Schemes, Lemma 01XT which has length \(1\) as an \(A\)-module, we conclude from the long exact cohomology sequence that (5) holds for \(n = -2\). And so on and so forth.

Lemma

Let \((A, \mathfrak m)\) be a regular local ring of dimension \(2\). Let \(f : X \to S = \Spec(A)\) be the blowing up of \(A\) in \(\mathfrak m\). Let \(\mathfrak m^n \subset I \subset \mathfrak m\) be an ideal. Let \(d \geq 0\) be the largest integer such that \[I \mathcal{O}_X \subset \mathcal{O}_X(-dE)\] where \(E\) is the exceptional divisor. Set \(\mathcal{I}' = I\mathcal{O}_X(dE) \subset \mathcal{O}_X\). Then \(d > 0\), the sheaf \(\mathcal{O}_X/\mathcal{I}'\) is supported in finitely many closed points \(x_1, \ldots, x_r\) of \(X\), and \[\begin{align*} \text{length}_A(A/I) & > \text{length}_A \Gamma(X, \mathcal{O}_X/\mathcal{I}') \\ & \geq \sum\nolimits_{i = 1, \ldots, r} \text{length}_{\mathcal{O}_{X, x_i}} (\mathcal{O}_{X, x_i}/\mathcal{I}'_{x_i}) \end{align*}\]

Proof

Since \(I \subset \mathfrak m\) we see that every element of \(I\) vanishes on \(E\). Thus we see that \(d \geq 1\). On the other hand, since \(\mathfrak m^n \subset I\) we see that \(d \leq n\). Consider the short exact sequence \[0 \to I\mathcal{O}_X \to \mathcal{O}_X \to \mathcal{O}_X/I\mathcal{O}_X \to 0\] Since \(I\mathcal{O}_X\) is globally generated, we see that \(H^1(X, I\mathcal{O}_X) = 0\) by Lemma 0AGS. Hence we obtain a surjection \(A/I \to \Gamma(X, \mathcal{O}_X/I\mathcal{O}_X)\). Consider the short exact sequence \[0 \to \mathcal{O}_X(-dE)/I\mathcal{O}_X \to \mathcal{O}_X/I\mathcal{O}_X \to \mathcal{O}_X/\mathcal{O}_X(-dE) \to 0\] By Divisors, Lemma 0AGB we see that \(\mathcal{O}_X(-dE)/I\mathcal{O}_X\) is supported in finitely many closed points of \(X\). In particular, this coherent sheaf has vanishing higher cohomology groups (detail omitted). Thus in the following diagram \[\xymatrix{ & & A/I \ar[d] \\ 0 \ar[r] & \Gamma(X, \mathcal{O}_X(-dE)/I\mathcal{O}_X) \ar[r] & \Gamma(X, \mathcal{O}_X/I\mathcal{O}_X) \ar[r] & \Gamma(X, \mathcal{O}_X/\mathcal{O}_X(-dE)) \ar[r] & 0 }\] the bottom row is exact and the vertical arrow surjective. We have \[\text{length}_A \Gamma(X, \mathcal{O}_X(-dE)/I\mathcal{O}_X) < \text{length}_A(A/I)\] since \(\Gamma(X, \mathcal{O}_X/\mathcal{O}_X(-dE))\) is nonzero. Namely, the image of \(1 \in \Gamma(X, \mathcal{O}_X)\) is nonzero as \(d > 0\).

To finish the proof we translate the results above into the statements of the lemma. Since \(\mathcal{O}_X(dE)\) is invertible we have \[\mathcal{O}_X/\mathcal{I}' = \mathcal{O}_X(-dE)/I\mathcal{O}_X \otimes_{\mathcal{O}_X} \mathcal{O}_X(dE).\] Thus \(\mathcal{O}_X/\mathcal{I}'\) and \(\mathcal{O}_X(-dE)/I\mathcal{O}_X\) are supported in the same set of finitely many closed points, say \(x_1, \ldots, x_r \in E \subset X\). Moreover we obtain \[\Gamma(X, \mathcal{O}_X(-dE)/I\mathcal{O}_X) = \bigoplus \mathcal{O}_X(-dE)_{x_i}/I\mathcal{O}_{X, x_i} \cong \bigoplus \mathcal{O}_{X, x_i}/\mathcal{I}'_{x_i} = \Gamma(X, \mathcal{O}_X/\mathcal{I}')\] because an invertible module over a local ring is trivial. Thus we obtain the strict inequality. We also get the second because \[\text{length}_A(\mathcal{O}_{X, x_i}/\mathcal{I}'_{x_i}) \geq \text{length}_{\mathcal{O}_{X, x_i}}(\mathcal{O}_{X, x_i}/\mathcal{I}'_{x_i})\] as is immediate from the definition of length.

Lemma

Let \((A, \mathfrak m, \kappa)\) be a regular local ring of dimension \(2\). Let \(f : X \to S = \Spec(A)\) be the blowing up of \(A\) in \(\mathfrak m\). Then \(\Omega_{X/S} = i_*\Omega_{E/\kappa}\), where \(i : E \to X\) is the immersion of the exceptional divisor.

Proof

Writing \(\mathbf{P}^1 = \mathbf{P}^1_S\), let \(r : X \to \mathbf{P}^1\) be as in Lemma 0AGQ. Then we have an exact sequence \[\mathcal{C}_{X/\mathbf{P}^1} \to r^*\Omega_{\mathbf{P}^1/S} \to \Omega_{X/S} \to 0\] see Morphisms, Lemma 01UZ. Since \(\Omega_{\mathbf{P}^1/S}|_E = \Omega_{E/\kappa}\) by Morphisms, Lemma 01V0 it suffices to see that the first arrow defines a surjection onto the kernel of the canonical map \(r^*\Omega_{\mathbf{P}^1/S} \to i_*\Omega_{E/\kappa}\). This we can do locally. With notation as in the proof of Lemma 0AGQ on an affine open of \(X\) the morphism \(f\) corresponds to the ring map \[A \to A[t]/(xt - y)\] where \(x, y \in \mathfrak m\) are generators. Thus \(\text{d}(xt - y) = x\text{d}t\) and \(y\text{d}t = t \cdot x \text{d}t\) which proves what we want.

Dominating by quadratic transformations

Using the result above we can prove that blowups in points dominate any modification of a regular \(2\) dimensional scheme.

Let \(X\) be a scheme. Let \(x \in X\) be a closed point. As usual, we view \(i : x = \Spec(\kappa(x)) \to X\) as a closed subscheme. The blowing up \(X' \to X\) of \(X\) at \(x\) is the blowing up of \(X\) in the closed subscheme \(x \subset X\). Observe that if \(X\) is locally Noetherian, then \(X' \to X\) is projective (in particular proper) by Divisors, Lemma 02NS.

Lemma

Let \(X\) be a Noetherian scheme. Let \(T \subset X\) be a finite set of closed points \(x\) such that \(\mathcal{O}_{X, x}\) is regular of dimension \(2\) for \(x \in T\). Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals such that \(\mathcal{O}_X/\mathcal{I}\) is supported on \(T\). Then there exists a sequence \[X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X\] where \(X_{i + 1} \to X_i\) is the blowing up of \(X_i\) at a closed point lying above a point of \(T\) such that \(\mathcal{I}\mathcal{O}_{X_n}\) is an invertible ideal sheaf.

Proof

Say \(T = \{x_1, \ldots, x_r\}\). Denote \(I_i\) the stalk of \(\mathcal{I}\) at \(x_i\). Set \[n_i = \text{length}_{\mathcal{O}_{X, x_i}}(\mathcal{O}_{X, x_i}/I_i)\] This is finite as \(\mathcal{O}_X/\mathcal{I}\) is supported on \(T\) and hence \(\mathcal{O}_{X, x_i}/I_i\) has support equal to \(\{\mathfrak m_{x_i}\}\) (see Algebra, Lemma 00L5). We are going to use induction on \(\sum n_i\). If \(n_i = 0\) for all \(i\), then \(\mathcal{I} = \mathcal{O}_X\) and we are done.

Suppose \(n_i > 0\). Let \(X' \to X\) be the blowing up of \(X\) in \(x_i\) (see discussion above the lemma). Since \(\Spec(\mathcal{O}_{X, x_i}) \to X\) is flat we see that \(X' \times_X \Spec(\mathcal{O}_{X, x_i})\) is the blowup of the ring \(\mathcal{O}_{X, x_i}\) in the maximal ideal, see Divisors, Lemma 0805. Hence the square in the commutative diagram \[\xymatrix{ \text{Proj}(\bigoplus\nolimits_{d \geq 0} \mathfrak m_{x_i}^d) \ar[r] \ar[d] & X' \ar[d] \\ \Spec(\mathcal{O}_{X, x_i}) \ar[r] & X }\] is cartesian. Let \(E \subset X'\) and \(E' \subset \text{Proj}(\bigoplus\nolimits_{d \geq 0} \mathfrak m_{x_i}^d)\) be the exceptional divisors. Let \(d \geq 1\) be the integer found in Lemma 0AGT for the ideal \(\mathcal{I}_i \subset \mathcal{O}_{X, x_i}\). Since the horizontal arrows in the diagram are flat, since \(E' \to E\) is surjective, and since \(E'\) is the pullback of \(E\), we see that \[\mathcal{I}\mathcal{O}_{X'} \subset \mathcal{O}_{X'}(-dE)\] (some details omitted). Set \(\mathcal{I}' = \mathcal{I}\mathcal{O}_{X'}(dE) \subset \mathcal{O}_{X'}\). Then we see that \(\mathcal{O}_{X'}/\mathcal{I}'\) is supported in finitely many closed points \(T' \subset |X'|\) because this holds over \(X \setminus \{x_i\}\) and for the pullback to \(\text{Proj}(\bigoplus\nolimits_{d \geq 0} \mathfrak m_{x_i}^d)\). The final assertion of Lemma 0AGT tells us that the sum of the lengths of the stalks \(\mathcal{O}_{X', x'}/\mathcal{I}'\mathcal{O}_{X', x'}\) for \(x'\) lying over \(x_i\) is \(< n_i\). Hence the sum of the lengths has decreased.

By induction hypothesis, there exists a sequence \[X'_n \to \ldots \to X'_1 \to X'\] of blowups at closed points lying over \(T'\) such that \(\mathcal{I}'\mathcal{O}_{X'_n}\) is invertible. Since \(\mathcal{I}'\mathcal{O}_{X'}(-dE) = \mathcal{I}\mathcal{O}_{X'}\), we see that \(\mathcal{I}\mathcal{O}_{X'_n} = \mathcal{I}'\mathcal{O}_{X'_n}(-d(f')^{-1}E)\) where \(f' : X'_n \to X'\) is the composition. Note that \((f')^{-1}E\) is an effective Cartier divisor by Divisors, Lemma 0809. Thus we are done by Divisors, Lemma 01WU.

Lemma

Let \(X\) be a Noetherian scheme. Let \(T \subset X\) be a finite set of closed points \(x\) such that \(\mathcal{O}_{X, x}\) is a regular local ring of dimension \(2\). Let \(f : Y \to X\) be a proper morphism of schemes which is an isomorphism over \(U = X \setminus T\). Then there exists a sequence \[X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X\] where \(X_{i + 1} \to X_i\) is the blowing up of \(X_i\) at a closed point \(x_i\) lying above a point of \(T\) and a factorization \(X_n \to Y \to X\) of the composition.

Proof

By More on Flatness, Lemma 081T there exists a \(U\)-admissible blowup \(X' \to X\) which dominates \(Y \to X\). Hence we may assume there exists an ideal sheaf \(\mathcal{I} \subset \mathcal{O}_X\) such that \(\mathcal{O}_X/\mathcal{I}\) is supported on \(T\) and such that \(Y\) is the blowing up of \(X\) in \(\mathcal{I}\). By Lemma 0AHH there exists a sequence \[X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X\] where \(X_{i + 1} \to X_i\) is the blowing up of \(X_i\) at a closed point \(x_i\) lying above a point of \(T\) such that \(\mathcal{I}\mathcal{O}_{X_n}\) is an invertible ideal sheaf. By the universal property of blowing up (Divisors, Lemma 0806) we find the desired factorization.

Lemma

Let \(S\) be a scheme. Let \(X\) be a scheme over \(S\) which is regular and has dimension \(2\). Let \(Y\) be a proper scheme over \(S\). Given an \(S\)-rational map \(f : U \to Y\) from \(X\) to \(Y\) there exists a sequence \[X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X\] and an \(S\)-morphism \(f_n : X_n \to Y\) such that \(X_{i + 1} \to X_i\) is the blowing up of \(X_i\) at a closed point not lying over \(U\) and \(f_n\) and \(f\) agree.

Proof

We may assume \(U\) contains every point of codimension \(1\), see Morphisms, Lemma 0BX7. Hence the complement \(T \subset X\) of \(U\) is a finite set of closed points whose local rings are regular of dimension \(2\). Applying Divisors, Lemma 0C4V we find a proper morphism \(p : X' \to X\) which is an isomorphism over \(U\) and a morphism \(f' : X' \to Y\) agreeing with \(f\) over \(U\). Apply Lemma 0AHI to the morphism \(p : X' \to X\). The composition \(X_n \to X' \to Y\) is the desired morphism.

Dominating by normalized blowups

In this section we prove that a modification of a surface can be dominated by a sequence of normalized blowups in points.

Definition

Let \(X\) be a scheme such that every quasi-compact open has finitely many irreducible components. Let \(x \in X\) be a closed point. The normalized blowup of \(X\) at \(x\) is the composition \(X'' \to X' \to X\) where \(X' \to X\) is the blowup of \(X\) in \(x\) and \(X'' \to X'\) is the normalization of \(X'\).

Here the normalization \(X'' \to X'\) is defined as the scheme \(X'\) has an open covering by opens which have finitely many irreducible components by Divisors, Lemma 0BFM. See Morphisms, Definition 035N for the definition of the normalization.

In general the normalized blowing up need not be proper even when \(X\) is Noetherian. Recall that a scheme is Nagata if it has an open covering by affines which are spectra of Nagata rings (Properties, Definition 033S).

Lemma

In Definition 0BBS if \(X\) is Nagata, then the normalized blowing up of \(X\) at \(x\) is normal, Nagata, and proper over \(X\).

Proof

The blowup morphism \(X' \to X\) is proper (as \(X\) is locally Noetherian we may apply Divisors, Lemma 02NS). Thus \(X'\) is Nagata (Morphisms, Lemma 035A). Therefore the normalization \(X'' \to X'\) is finite (Morphisms, Lemma 035S) and we conclude that \(X'' \to X\) is proper as well (Morphisms, Lemmas 01WN and 01W3). It follows that the normalized blowing up is a normal (Morphisms, Lemma 035Q) Nagata algebraic space.

In the following lemma we need to assume \(X\) is Noetherian in order to make sure that it has finitely many irreducible components. Then the properness of \(f : Y \to X\) assures that \(Y\) has finitely many irreducible components too and it makes sense to require \(f\) to be birational (Morphisms, Definition 01RO).

Lemma

Let \(X\) be a scheme which is Noetherian, Nagata, and has dimension \(2\). Let \(f : Y \to X\) be a proper birational morphism. Then there exists a commutative diagram \[\xymatrix{ X_n \ar[r] \ar[d] & X_{n - 1} \ar[r] & \ldots \ar[r] & X_1 \ar[r] & X_0 \ar[d] \\ Y \ar[rrrr] & & & & X }\] where \(X_0 \to X\) is the normalization and where \(X_{i + 1} \to X_i\) is the normalized blowing up of \(X_i\) at a closed point.

Proof

We will use the results of Morphisms, Sections 0359, 02JT, and 035E without further mention. We may replace \(Y\) by its normalization. Let \(X_0 \to X\) be the normalization. The morphism \(Y \to X\) factors through \(X_0\). Thus we may assume that both \(X\) and \(Y\) are normal.

Assume \(X\) and \(Y\) are normal. The morphism \(f : Y \to X\) is an isomorphism over an open which contains every point of codimension \(0\) and \(1\) in \(Y\) and every point of \(Y\) over which the fibre is finite, see Varieties, Lemma 0BFP. Hence there is a finite set of closed points \(T \subset X\) such that \(f\) is an isomorphism over \(X \setminus T\). For each \(x \in T\) the fibre \(Y_x\) is a proper geometrically connected scheme of dimension \(1\) over \(\kappa(x)\), see More on Morphisms, Lemma 0AY8. Thus \[BadCurves(f) = \{C \subset Y\text{ closed} \mid \dim(C) = 1, f(C) = \text{a point}\}\] is a finite set. We will prove the lemma by induction on the number of elements of \(BadCurves(f)\). The base case is the case where \(BadCurves(f)\) is empty, and in that case \(f\) is an isomorphism.

Fix \(x \in T\). Let \(X' \to X\) be the normalized blowup of \(X\) at \(x\) and let \(Y'\) be the normalization of \(Y \times_X X'\). Picture \[\xymatrix{ Y' \ar[r]_{f'} \ar[d] & X' \ar[d] \\ Y \ar[r]^f & X }\] Let \(x' \in X'\) be a closed point lying over \(x\) such that the fibre \(Y'_{x'}\) has dimension \(\geq 1\). Let \(C' \subset Y'\) be an irreducible component of \(Y'_{x'}\), i.e., \(C' \in BadCurves(f')\). Since \(Y' \to Y \times_X X'\) is finite we see that \(C'\) must map to an irreducible component \(C \subset Y_x\). If is clear that \(C \in BadCurves(f)\). Since \(Y' \to Y\) is birational and hence an isomorphism over points of codimension \(1\) in \(Y\), we see that we obtain an injective map \[BadCurves(f') \longrightarrow BadCurves(f)\] Thus it suffices to show that after a finite number of these normalized blowups we get rid at of at least one of the bad curves, i.e., the displayed map is not surjective.

We will get rid of a bad curve using an argument due to Zariski. Pick \(C \in BadCurves(f)\) lying over our \(x\). Denote \(\mathcal{O}_{Y, C}\) the local ring of \(Y\) at the generic point of \(C\). Choose an element \(u \in \mathcal{O}_{X, C}\) whose image in the residue field \(R(C)\) is transcendental over \(\kappa(x)\) (we can do this because \(R(C)\) has transcendence degree \(1\) over \(\kappa(x)\) by Varieties, Lemma 0A21). We can write \(u = a/b\) with \(a, b \in \mathcal{O}_{X, x}\) as \(\mathcal{O}_{Y, C}\) and \(\mathcal{O}_{X, x}\) have the same fraction fields. By our choice of \(u\) it must be the case that \(a, b \in \mathfrak m_x\). Hence \[N_{u, a, b} = \min \{\text{ord}_{\mathcal{O}_{Y, C}}(a), \text{ord}_{\mathcal{O}_{Y, C}}(b)\} > 0\] Thus we can do descending induction on this integer. Let \(X' \to X\) be the normalized blowing up of \(x\) and let \(Y'\) be the normalization of \(X' \times_X Y\) as above. We will show that if \(C\) is the image of some bad curve \(C' \subset Y'\) lying over \(x' \in X'\), then there exists a choice of \(a', b' \mathcal{O}_{X', x'}\) such that \(N_{u, a', b'} < N_{u, a, b}\). This will finish the proof. Namely, since \(X' \to X\) factors through the blowing up, we see that there exists a nonzero element \(d \in \mathfrak m_{x'}\) such that \(a = a' d\) and \(b = b' d\) (namely, take \(d\) to be the local equation for the exceptional divisor of the blowup). Since \(Y' \to Y\) is an isomorphism over an open containing the generic point of \(C\) (seen above) we see that \(\mathcal{O}_{Y', C'} = \mathcal{O}_{Y, C}\). Hence \[\text{ord}_{\mathcal{O}_{Y, C}}(a) = \text{ord}_{\mathcal{O}_{Y', C'}}(a' d) = \text{ord}_{\mathcal{O}_{Y', C'}}(a') + \text{ord}_{\mathcal{O}_{Y', C'}}(d) > \text{ord}_{\mathcal{O}_{Y', C'}}(a')\] Similarly for \(b\) and the proof is complete.

Lemma

Let \(S\) be a scheme. Let \(X\) be a scheme over \(S\) which is Noetherian, Nagata, and has dimension \(2\). Let \(Y\) be a proper scheme over \(S\). Given an \(S\)-rational map \(f : U \to Y\) from \(X\) to \(Y\) there exists a sequence \[X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 \to X\] and an \(S\)-morphism \(f_n : X_n \to Y\) such that \(X_0 \to X\) is the normalization, \(X_{i + 1} \to X_i\) is the normalized blowing up of \(X_i\) at a closed point, and \(f_n\) and \(f\) agree.

Proof

Applying Divisors, Lemma 0C4V we find a proper morphism \(p : X' \to X\) which is an isomorphism over \(U\) and a morphism \(f' : X' \to Y\) agreeing with \(f\) over \(U\). Apply Lemma 0BBT to the morphism \(p : X' \to X\). The composition \(X_n \to X' \to Y\) is the desired morphism.

Modifying over local rings

Let \(S\) be a scheme. Let \(s_1, \ldots, s_n \in S\) be pairwise distinct closed points. Assume that the open embedding \[U = S \setminus \{s_1, \ldots, s_n\} \longrightarrow S\] is quasi-compact. Denote \(FP_{S, \{s_1, \ldots, s_n\}}\) the category of morphisms \(f : X \to S\) of finite presentation which induce an isomorphism \(f^{-1}(U) \to U\). Morphisms are morphisms of schemes over \(S\). For each \(i\) set \(S_i = \Spec(\mathcal{O}_{S, s_i})\) and let \(V_i = S_i \setminus \{s_i\}\). Denote \(FP_{S_i, s_i}\) the category of morphisms \(g_i : Y_i \to S_i\) of finite presentation which induce an isomorphism \(g_i^{-1}(V_i) \to V_i\). Morphisms are morphisms over \(S_i\). Base change defines an functor [0BFU]\[\begin{equation} F : FP_{S, \{s_1, \ldots, s_n\}} \longrightarrow FP_{S_1, s_1} \times \ldots \times FP_{S_n, s_n} \end{equation}\] To reduce at least some of the problems in this chapter to the case of local rings we have the following lemma.

Lemma

The functor \(F\) (0BFU) is an equivalence.

Proof

For \(n = 1\) this is Limits, Lemma 0B3X. For \(n > 1\) the lemma can be proved in exactly the same way or it can be deduced from it. For example, suppose that \(g_i : Y_i \to S_i\) are objects of \(FP_{S_i, s_i}\). Then by the case \(n = 1\) we can find \(f'_i : X'_i \to S\) of finite presentation which are isomorphisms over \(S \setminus \{s_i\}\) and whose base change to \(S_i\) is \(g_i\). Then we can set \[f : X = X'_1 \times_S \ldots \times_S X'_n \to S\] This is an object of \(FP_{S, \{s_1, \ldots, s_n\}}\) whose base change by \(S_i \to S\) recovers \(g_i\). Thus the functor is essentially surjective. We omit the proof of fully faithfulness.

Lemma

Let \(S, s_i, S_i\) be as in (0BFU). If \(f : X \to S\) corresponds to \(g_i : Y_i \to S_i\) under \(F\), then \(f\) is separated, proper, finite, if and only if \(g_i\) is so for \(i = 1, \ldots, n\).

Proof

Follows from Limits, Lemma 0BFN.

Lemma

Let \(S, s_i, S_i\) be as in (0BFU). If \(f : X \to S\) corresponds to \(g_i : Y_i \to S_i\) under \(F\), then \(X_{s_i} \cong (Y_i)_{s_i}\) as schemes over \(\kappa(s_i)\).

Proof

This is clear.

Lemma

Let \(S, s_i, S_i\) be as in (0BFU) and assume \(f : X \to S\) corresponds to \(g_i : Y_i \to S_i\) under \(F\). Then there exists a factorization \[X = Z_m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = S\] of \(f\) where \(Z_{j + 1} \to Z_j\) is the blowing up of \(Z_j\) at a closed point \(z_j\) lying over \(\{s_1, \ldots, s_n\}\) if and only if for each \(i\) there exists a factorization \[Y_i = Z_{i, m_i} \to Z_{i, m_i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = S_i\] of \(g_i\) where \(Z_{i, j + 1} \to Z_{i, j}\) is the blowing up of \(Z_{i, j}\) at a closed point \(z_{i, j}\) lying over \(s_i\).

Proof

Let’s start with a sequence of blowups \(Z_m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = S\). The first morphism \(Z_1 \to S\) is given by blowing up one of the \(s_i\), say \(s_1\). Applying \(F\) to \(Z_1 \to S\) we find a blowup \(Z_{1, 1} \to S_1\) at \(s_1\) is the blowing up at \(s_1\) and otherwise \(Z_{i, 0} = S_i\) for \(i > 1\). In the next step, we either blow up one of the \(s_i\), \(i \geq 2\) on \(Z_1\) or we pick a closed point \(z_1\) of the fibre of \(Z_1 \to S\) over \(s_1\). In the first case it is clear what to do and in the second case we use that \((Z_1)_{s_1} \cong (Z_{1, 1})_{s_1}\) (Lemma 0BFX) to get a closed point \(z_{1, 1} \in Z_{1, 1}\) corresponding to \(z_1\). Then we set \(Z_{1, 2} \to Z_{1, 1}\) equal to the blowing up in \(z_{1, 1}\). Continuing in this manner we construct the factorizations of each \(g_i\).

Conversely, given sequences of blowups \(Z_{i, m_i} \to Z_{i, m_i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = S_i\) we construct the sequence of blowing ups of \(S\) in exactly the same manner.

Here is the analogue of Lemma 0BFY for normalized blowups.

Lemma

Let \(S, s_i, S_i\) be as in (0BFU) and assume \(f : X \to S\) corresponds to \(g_i : Y_i \to S_i\) under \(F\). Assume every quasi-compact open of \(S\) has finitely many irreducible components. Then there exists a factorization \[X = Z_m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = S\] of \(f\) where \(Z_{j + 1} \to Z_j\) is the normalized blowing up of \(Z_j\) at a closed point \(z_j\) lying over \(\{x_1, \ldots, x_n\}\) if and only if for each \(i\) there exists a factorization \[Y_i = Z_{i, m_i} \to Z_{i, m_i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = S_i\] of \(g_i\) where \(Z_{i, j + 1} \to Z_{i, j}\) is the normalized blowing up of \(Z_{i, j}\) at a closed point \(z_{i, j}\) lying over \(s_i\).

Proof

The assumption on \(S\) is used to assure us (successively) that the schemes we are normalizing have locally finitely many irreducible components so that the statement makes sense. Having said this the lemma follows by the exact same argument as used to prove Lemma 0BFY.

Vanishing

In this section we will often work in the following setting. Recall that a modification is a proper birational morphism between integral schemes (Morphisms, Definition 0AAZ).

Situation

Here \((A, \mathfrak m, \kappa)\) be a local Noetherian normal domain of dimension \(2\). Let \(s\) be the closed point of \(S = \Spec(A)\) and \(U = S \setminus \{s\}\). Let \(f : X \to S\) be a modification. We denote \(C_1, \ldots, C_r\) the irreducible components of the special fibre \(X_s\) of \(f\).

By Varieties, Lemma 0BFP the morphism \(f\) defines an isomorphism \(f^{-1}(U) \to U\). The special fibre \(X_s\) is proper over \(\Spec(\kappa)\) and has dimension at most \(1\) by Varieties, Lemma 0B2K. By Stein factorization (More on Morphisms, Lemma 0AY8) we have \(f_*\mathcal{O}_X = \mathcal{O}_S\) and the special fibre \(X_s\) is geometrically connected over \(\kappa\). If \(X_s\) has dimension \(0\), then \(f\) is finite (More on Morphisms, Lemma 02UP) and hence an isomorphism (Morphisms, Lemma 0AB1). We will discard this uninteresting case and we conclude that \(\dim(C_i) = 1\) for \(i = 1, \ldots, r\).

Lemma

In Situation 0AX8 there exists a \(U\)-admissible blowup \(X' \to S\) which dominates \(X\).

Proof

This is a special case of More on Flatness, Lemma 081T.

Lemma

In Situation 0AX8 there exists a nonzero \(f \in \mathfrak m\) such that for every \(i = 1, \ldots, r\) there exist

1. a closed point \(x_i \in C_i\) with \(x_i \not \in C_j\) for \(j \not = i\),

2. a factorization \(f = g_i f_i\) of \(f\) in \(\mathcal{O}_{X, x_i}\) such that \(g_i \in \mathfrak m_{x_i}\) maps to a nonzero element of \(\mathcal{O}_{C_i, x_i}\).

Proof

We will use the observations made following Situation 0AX8 without further mention. Pick a closed point \(x_i \in C_i\) which is not in \(C_j\) for \(j \not = i\). Pick \(g_i \in \mathfrak m_{x_i}\) which maps to a nonzero element of \(\mathcal{O}_{C_i, x_i}\). Since the fraction field of \(A\) is the fraction field of \(\mathcal{O}_{X_i, x_i}\) we can write \(g_i = a_i/b_i\) for some \(a_i, b_i \in A\). Take \(f = \prod a_i\).

Lemma

In Situation 0AX8 assume \(X\) is normal. Let \(Z \subset X\) be a nonempty effective Cartier divisor such that \(Z \subset X_s\) set theoretically. Then the conormal sheaf of \(Z\) is not trivial. More precisely, there exists an \(i\) such that \(C_i \subset Z\) and \(\deg(\mathcal{C}_{Z/X}|_{C_i}) > 0\).

Proof

We will use the observations made following Situation 0AX8 without further mention. Let \(f\) be a function as in Lemma 0AX9. Let \(\xi_i \in C_i\) be the generic point. Let \(\mathcal{O}_i\) be the local ring of \(X\) at \(\xi_i\). Then \(\mathcal{O}_i\) is a discrete valuation ring. Let \(e_i\) be the valuation of \(f\) in \(\mathcal{O}_i\), so \(e_i > 0\). Let \(h_i \in \mathcal{O}_i\) be a local equation for \(Z\) and let \(d_i\) be its valuation. Then \(d_i \geq 0\). Choose and fix \(i\) with \(d_i/e_i\) maximal (then \(d_i > 0\) as \(Z\) is not empty). Replace \(f\) by \(f^{d_i}\) and \(Z\) by \(e_iZ\). This is permissible, by the relation \(\mathcal{O}_X(e_i Z) = \mathcal{O}_X(Z)^{\otimes e_i}\), the relation between the conormal sheaf and \(\mathcal{O}_X(Z)\) (see Divisors, Lemmas 02OP and 0B3P, and since the degree gets multiplied by \(e_i\), see Varieties, Lemma 0AYX. Let \(\mathcal{I}\) be the ideal sheaf of \(Z\) so that \(\mathcal{C}_{Z/X} = \mathcal{I}|_Z\). Consider the image \(\overline{f}\) of \(f\) in \(\Gamma(Z, \mathcal{O}_Z)\). By our choices above we see that \(\overline{f}\) vanishes in the generic points of irreducible components of \(Z\) (these are all generic points of \(C_j\) as \(Z\) is contained in the special fibre). On the other hand, \(Z\) is \((S_1)\) by Divisors, Lemma 0B3S. Thus the scheme \(Z\) has no embedded associated points and we conclude that \(\overline{f} = 0\) (Divisors, Lemmas 0346 and 0B3M). Hence \(f\) is a global section of \(\mathcal{I}\) which generates \(\mathcal{I}_{\xi_i}\) by construction. Thus the image \(s_i\) of \(f\) in \(\Gamma(C_i, \mathcal{I}|_{C_i})\) is nonzero. However, our choice of \(f\) guarantees that \(s_i\) has a zero at \(x_i\). Hence the degree of \(\mathcal{I}|_{C_i}\) is \(> 0\) by Varieties, Lemma 0B40.

Lemma

In Situation 0AX8 assume \(X\) is normal and \(A\) Nagata. The map \[H^1(X, \mathcal{O}_X) \longrightarrow H^1(f^{-1}(U), \mathcal{O}_X)\] is injective.

Proof

Let \(0 \to \mathcal{O}_X \to \mathcal{E} \to \mathcal{O}_X \to 0\) be the extension corresponding to a nontrivial element \(\xi\) of \(H^1(X, \mathcal{O}_X)\) (Cohomology, Lemma 0B3A). Let \(\pi : P = \mathbf{P}(\mathcal{E}) \to X\) be the projective bundle associated to \(\mathcal{E}\). The surjection \(\mathcal{E} \to \mathcal{O}_X\) defines a section \(\sigma : X \to P\) whose conormal sheaf is isomorphic to \(\mathcal{O}_X\) (Divisors, Lemma 0B3V). If the restriction of \(\xi\) to \(f^{-1}(U)\) is trivial, then we get a map \(\mathcal{E}|_{f^{-1}(U)} \to \mathcal{O}_{f^{-1}(U)}\) splitting the injection \(\mathcal{O}_X \to \mathcal{E}\). This defines a second section \(\sigma' : f^{-1}(U) \to P\) disjoint from \(\sigma\). Since \(\xi\) is nontrivial we conclude that \(\sigma'\) cannot extend to all of \(X\) and be disjoint from \(\sigma\). Let \(X' \subset P\) be the scheme theoretic image of \(\sigma'\) (Morphisms, Definition 01R7). Picture \[\xymatrix{ & X' \ar[rd]_g \ar[r] & P \ar[d]_\pi \\ f^{-1}(U) \ar[ru]_{\sigma'} \ar[rr] & & X \ar@/_/[u]_\sigma }\] The morphism \(P \setminus \sigma(X) \to X\) is affine. If \(X' \cap \sigma(X) = \emptyset\), then \(X' \to X\) is both affine and proper, hence finite (Morphisms, Lemma 01WN), hence an isomorphism (as \(X\) is normal, see Morphisms, Lemma 0AB1). This is impossible as mentioned above.

Let \(X^\nu\) be the normalization of \(X'\). Since \(A\) is Nagata, we see that \(X^\nu \to X'\) is finite (Morphisms, Lemmas 035S and 035B). Let \(Z \subset X^\nu\) be the pullback of the effective Cartier divisor \(\sigma(X) \subset P\). By the above we see that \(Z\) is not empty and is contained in the closed fibre of \(X^\nu \to S\). Since \(P \to X\) is smooth, we see that \(\sigma(X)\) is an effective Cartier divisor (Divisors, Lemma 067R). Hence \(Z \subset X^\nu\) is an effective Cartier divisor too. Since the conormal sheaf of \(\sigma(X)\) in \(P\) is \(\mathcal{O}_X\), the conormal sheaf of \(Z\) in \(X^\nu\) (which is a priori invertible) is \(\mathcal{O}_Z\) by Morphisms, Lemma 0473. This is impossible by Lemma 0AXA and the proof is complete.

Lemma

In Situation 0AX8 assume \(X\) is normal and \(A\) Nagata. Then \[\Hom_{D(A)}(\kappa[-1], Rf_*\mathcal{O}_X)\] is zero. This uses \(D(A) = D_\QCoh(\mathcal{O}_S)\) to think of \(Rf_*\mathcal{O}_X\) as an object of \(D(A)\).

Proof

By adjointness of \(Rf_*\) and \(Lf^*\) such a map is the same thing as a map \(\alpha : Lf^*\kappa[-1] \to \mathcal{O}_X\). Note that \[H^i(Lf^*\kappa[-1]) = \left\{ \begin{matrix} 0 & \text{if} & i > 1 \\ \mathcal{O}_{X_s} & \text{if} & i = 1 \\ \text{some }\mathcal{O}_{X_s}\text{-module} & \text{if} & i \leq 0 \end{matrix} \right.\] Since \(\Hom(H^0(Lf^*\kappa[-1]), \mathcal{O}_X) = 0\) as \(\mathcal{O}_X\) is torsion free, the spectral sequence for \(\Ext\) (Cohomology on Sites, Example 07AA) implies that \(\Hom_{D(\mathcal{O}_X)}(Lf^*\kappa[-1], \mathcal{O}_X)\) is equal to \(\Ext^1_{\mathcal{O}_X}(\mathcal{O}_{X_s}, \mathcal{O}_X)\). We conclude that \(\alpha : Lf^*\kappa[-1] \to \mathcal{O}_X\) is given by an extension \[0 \to \mathcal{O}_X \to \mathcal{E} \to \mathcal{O}_{X_s} \to 0\] By Lemma 0AXB the pullback of this extension via the surjection \(\mathcal{O}_X \to \mathcal{O}_{X_s}\) is zero (since this pullback is clearly split over \(f^{-1}(U)\)). Thus \(1 \in \mathcal{O}_{X_s}\) lifts to a global section \(s\) of \(\mathcal{E}\). Multiplying \(s\) by the ideal sheaf \(\mathcal{I}\) of \(X_s\) we obtain an \(\mathcal{O}_X\)-module map \(c_s : \mathcal{I} \to \mathcal{O}_X\). Applying \(f_*\) we obtain an \(A\)-linear map \(f_*c_s : \mathfrak m \to A\). Since \(A\) is a Noetherian normal local domain this map is given by multiplication by an element \(a \in A\). Changing \(s\) into \(s - a\) we find that \(s\) is annihilated by \(\mathcal{I}\) and the extension is trivial as desired.

Remark

Let \(X\) be an integral Noetherian normal scheme of dimension \(2\). In this case the following are equivalent

1. \(X\) has a dualizing complex \(\omega_X^\bullet\),

2. there is a coherent \(\mathcal{O}_X\)-module \(\omega_X\) such that \(\omega_X[n]\) is a dualizing complex, where \(n\) can be any integer.

This follows from the fact that \(X\) is Cohen-Macaulay (Properties, Lemma 0B3D) and Duality for Schemes, Lemma 0AWT. In this situation we will say that \(\omega_X\) is a dualizing module in accordance with Duality for Schemes, Section 0AWH. In particular, when \(A\) is a Noetherian normal local domain of dimension \(2\), then we say \(A\) has a dualizing module \(\omega_A\) if the above is true. In this case, if \(X \to \Spec(A)\) is a normal modification, then \(X\) has a dualizing module too, see Duality for Schemes, Example 0AWI. In this situation we always denote \(\omega_X\) the dualizing module normalized with respect to \(\omega_A\), i.e., such that \(\omega_X[2]\) is the dualizing complex normalized relative to \(\omega_A[2]\). See Duality for Schemes, Section 0AU5.

The Grauert-Riemenschneider vanishing of the next proposition is a formal consequence of Lemma 0AXC and the general theory of duality.

Proposition

In Situation 0AX8 assume

1. \(X\) is a normal scheme,

2. \(A\) is Nagata and has a dualizing complex \(\omega_A^\bullet\).

Let \(\omega_X\) be the dualizing module of \(X\) (Remark 0B4R). Then \(R^1f_*\omega_X = 0\).

Proof

In this proof we will use the identification \(D(A) = D_\QCoh(\mathcal{O}_S)\) to identify quasi-coherent \(\mathcal{O}_S\)-modules with \(A\)-modules. Moreover, we may assume that \(\omega_A^\bullet\) is normalized, see Dualizing Complexes, Section 0A7M. Since \(X\) is a Noetherian normal \(2\)-dimensional scheme it is Cohen-Macaulay (Properties, Lemma 0B3D). Thus \(\omega_X^\bullet = \omega_X[2]\) (Duality for Schemes, Lemma 0AWT and the normalization in Duality for Schemes, Example 0AWI). If the proposition is false, then we can find a nonzero map \(R^1f_*\omega_X \to \kappa\). In other words we obtain a nonzero map \(\alpha : Rf_*\omega_X^\bullet \to \kappa[1]\). Applying \(R\Hom_A(-, \omega_A^\bullet)\) we get a nonzero map \[\beta : \kappa[-1] \longrightarrow Rf_*\mathcal{O}_X\] which is impossible by Lemma 0AXC. To see that \(R\Hom_A(-, \omega_A^\bullet)\) does what we said, first note that \[R\Hom_A(\kappa[1], \omega_A^\bullet) = R\Hom_A(\kappa, \omega_A^\bullet)[-1] = \kappa[-1]\] as \(\omega_A^\bullet\) is normalized and we have \[R\Hom_A(Rf_*\omega_X^\bullet, \omega_A^\bullet) = Rf_*R\SheafHom_{\mathcal{O}_X}(\omega_X^\bullet, \omega_X^\bullet) = Rf_*\mathcal{O}_X\] The first equality by Duality for Schemes, Example 0GEW and the fact that \(\omega_X^\bullet = f^!\omega_A^\bullet\) by construction, and the second equality because \(\omega_X^\bullet\) is a dualizing complex for \(X\) (which goes back to Duality for Schemes, Lemma 0AA3).

Boundedness

In this section we begin the discussion which will lead to a reduction to the case of rational singularities for \(2\)-dimensional schemes.

Lemma

Let \((A, \mathfrak m, \kappa)\) be a Noetherian normal local domain of dimension \(2\). Consider a commutative diagram \[\xymatrix{ X' \ar[rd]_{f'} \ar[rr]_g & & X \ar[ld]^f \\ & \Spec(A) }\] where \(f\) and \(f'\) are modifications as in Situation 0AX8 and \(X\) normal. Then we have a short exact sequence \[0 \to H^1(X, \mathcal{O}_X) \to H^1(X', \mathcal{O}_{X'}) \to H^0(X, R^1g_*\mathcal{O}_{X'}) \to 0\] Also \(\dim(\text{Supp}(R^1g_*\mathcal{O}_{X'})) = 0\) and \(R^1g_*\mathcal{O}_{X'}\) is generated by global sections.

Proof

We will use the observations made following Situation 0AX8 without further mention. As \(X\) is normal and \(g\) is dominant and birational, we have \(g_*\mathcal{O}_{X'} = \mathcal{O}_X\), see for example More on Morphisms, Lemma 0AY8. Since the fibres of \(g\) have dimension \(\leq 1\), we have \(R^pg_*\mathcal{O}_{X'} = 0\) for \(p > 1\), see for example Cohomology of Schemes, Lemma 02V7. The support of \(R^1g_*\mathcal{O}_{X'}\) is contained in the set of points of \(X\) where the fibres of \(g'\) have dimension \(\geq 1\). Thus it is contained in the set of images of those irreducible components \(C' \subset X'_s\) which map to points of \(X_s\) which is a finite set of closed points (recall that \(X'_s \to X_s\) is a morphism of proper \(1\)-dimensional schemes over \(\kappa\)). Then \(R^1g_*\mathcal{O}_{X'}\) is globally generated by Cohomology of Schemes, Lemma 0B3J. Using the morphism \(f : X \to S\) and the references above we find that \(H^p(X, \mathcal{F}) = 0\) for \(p > 1\) for any coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\). Hence the short exact sequence of the lemma is a consequence of the Leray spectral sequence for \(g\) and \(\mathcal{O}_{X'}\), see Cohomology, Lemma 01F2.

Lemma

Let \((A, \mathfrak m, \kappa)\) be a local normal Nagata domain of dimension \(2\). Let \(a \in A\) be nonzero. There exists an integer \(N\) such that for every modification \(f : X \to \Spec(A)\) with \(X\) normal the \(A\)-module \[M_{X, a} = \Coker(A \longrightarrow H^0(Z, \mathcal{O}_Z))\] where \(Z \subset X\) is cut out by \(a\) has length bounded by \(N\).

Proof

By the short exact sequence \(0 \to \mathcal{O}_X \xrightarrow{a} \mathcal{O}_X \to \mathcal{O}_Z \to 0\) we see that [0AXK]\[\begin{equation} M_{X, a} = H^1(X, \mathcal{O}_X)[a] \end{equation}\] Here \(N[a] = \{n \in N \mid an = 0\}\) for an \(A\)-module \(N\). Thus if \(a\) divides \(b\), then \(M_{X, a} \subset M_{X, b}\). Suppose that for some \(c \in A\) the modules \(M_{X, c}\) have bounded length. Then for every \(X\) we have an exact sequence \[0 \to M_{X, c} \to M_{X, c^2} \to M_{X, c}\] where the second arrow is given by multiplication by \(c\). Hence we see that \(M_{X, c^2}\) has bounded length as well. Thus it suffices to find a \(c \in A\) for which the lemma is true such that \(a\) divides \(c^n\) for some \(n > 0\). By More on Algebra, Lemma 0AXI we may assume \(A/(a)\) is a reduced ring.

Assume that \(A/(a)\) is reduced. Let \(A/(a) \subset B\) be the normalization of \(A/(a)\) in its quotient ring. Because \(A\) is Nagata, we see that \(\Coker(A \to B)\) is finite. We claim the length of this finite module is a bound. To see this, consider \(f : X \to \Spec(A)\) as in the lemma and let \(Z' \subset Z\) be the scheme theoretic closure of \(Z \cap f^{-1}(U)\). Then \(Z' \to \Spec(A/(a))\) is finite for example by Varieties, Lemma 0AB7. Hence \(Z' = \Spec(B')\) with \(A/(a) \subset B' \subset B\). On the other hand, we claim the map \[H^0(Z, \mathcal{O}_Z) \to H^0(Z', \mathcal{O}_{Z'})\] is injective. Namely, if \(s \in H^0(Z, \mathcal{O}_Z)\) is in the kernel, then the restriction of \(s\) to \(f^{-1}(U) \cap Z\) is zero. Hence the image of \(s\) in \(H^1(X, \mathcal{O}_X)\) vanishes in \(H^1(f^{-1}(U), \mathcal{O}_X)\). By Lemma 0AXB we see that \(s\) comes from an element \(\tilde s\) of \(A\). But by assumption \(\tilde s\) maps to zero in \(B'\) which implies that \(s = 0\). Putting everything together we see that \(M_{X, a}\) is a subquotient of \(B'/A\), namely not every element of \(B'\) extends to a global section of \(\mathcal{O}_Z\), but in any case the length of \(M_{X, a}\) is bounded by the length of \(B/A\).

In some cases, resolution of singularities reduces to the case of rational singularities.

Definition

Let \((A, \mathfrak m, \kappa)\) be a local normal Nagata domain of dimension \(2\).

1. We say \(A\) defines a rational singularity if for every normal modification \(X \to \Spec(A)\) we have \(H^1(X, \mathcal{O}_X) = 0\).

2. We say that reduction to rational singularities is possible for \(A\) if the length of the \(A\)-modules \[H^1(X, \mathcal{O}_X)\] is bounded for all modifications \(X \to \Spec(A)\) with \(X\) normal.

The meaning of the language in (2) is explained by Lemma 0B4P. The following lemma says roughly speaking that local rings of modifications of \(\Spec(A)\) with \(A\) defining a rational singularity also define rational singularities.

Lemma

Let \((A, \mathfrak m, \kappa)\) be a local normal Nagata domain of dimension \(2\) which defines a rational singularity. Let \(A \subset B\) be a local extension of domains with the same fraction field which is essentially of finite type such that \(\dim(B) = 2\) and \(B\) normal. Then \(B\) defines a rational singularity.

Proof

Choose a finite type \(A\)-algebra \(C\) such that \(B = C_\mathfrak q\) for some prime \(\mathfrak q \subset C\). After replacing \(C\) by the image of \(C\) in \(B\) we may assume that \(C\) is a domain with fraction field equal to the fraction field of \(A\). Then we can choose a closed immersion \(\Spec(C) \to \mathbf{A}^n_A\) and take the closure in \(\mathbf{P}^n_A\) to conclude that \(B\) is isomorphic to \(\mathcal{O}_{X, x}\) for some closed point \(x \in X\) of a projective modification \(X \to \Spec(A)\). (Morphisms, Lemma 02JU, shows that \(\kappa(x)\) is finite over \(\kappa\) and then Morphisms, Lemma 01TE shows that \(x\) is a closed point.) Let \(\nu : X^\nu \to X\) be the normalization. Since \(A\) is Nagata the morphism \(\nu\) is finite (Morphisms, Lemma 035S). Thus \(X^\nu\) is projective over \(A\) by More on Morphisms, Lemma 0B46. Since \(B = \mathcal{O}_{X, x}\) is normal, we see that \(\mathcal{O}_{X, x} = (\nu_*\mathcal{O}_{X^\nu})_x\). Hence there is a unique point \(x^\nu \in X^\nu\) lying over \(x\) and \(\mathcal{O}_{X^\nu, x^\nu} = \mathcal{O}_{X, x}\). Thus we may assume \(X\) is normal and projective over \(A\). Let \(Y \to \Spec(\mathcal{O}_{X, x}) = \Spec(B)\) be a modification with \(Y\) normal. We have to show that \(H^1(Y, \mathcal{O}_Y) = 0\). By Limits, Lemma 0B3X we can find a morphism of schemes \(g : X' \to X\) which is an isomorphism over \(X \setminus \{x\}\) such that \(X' \times_X \Spec(\mathcal{O}_{X, x})\) is isomorphic to \(Y\). Then \(g\) is a modification as it is proper by Limits, Lemma 0BFN. The local ring of \(X'\) at a point of \(x'\) is either isomorphic to the local ring of \(X\) at \(g(x')\) if \(g(x') \not = x\) and if \(g(x') = x\), then the local ring of \(X'\) at \(x'\) is isomorphic to the local ring of \(Y\) at the corresponding point. Hence we see that \(X'\) is normal as both \(X\) and \(Y\) are normal. Thus \(H^1(X', \mathcal{O}_{X'}) = 0\) by our assumption on \(A\). By Lemma 0AXF we have \(R^1g_*\mathcal{O}_{X'} = 0\). Clearly this means that \(H^1(Y, \mathcal{O}_Y) = 0\) as desired.

Lemma

Let \((A, \mathfrak m, \kappa)\) be a local normal Nagata domain of dimension \(2\). If reduction to rational singularities is possible for \(A\), then there exists a finite sequence of normalized blowups \[X = X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = \Spec(A)\] in closed points such that for any closed point \(x \in X\) the local ring \(\mathcal{O}_{X, x}\) defines a rational singularity. In particular \(X \to \Spec(A)\) is a modification and \(X\) is a normal scheme projective over \(A\).

Proof

We choose a modification \(X \to \Spec(A)\) with \(X\) normal which maximizes the length of \(H^1(X, \mathcal{O}_X)\). By Lemma 0AXF for any further modification \(g : X' \to X\) with \(X'\) normal we have \(R^1g_*\mathcal{O}_{X'} = 0\) and \(H^1(X, \mathcal{O}_X) = H^1(X', \mathcal{O}_{X'})\).

Let \(x \in X\) be a closed point. We will show that \(\mathcal{O}_{X, x}\) defines a rational singularity. Let \(Y \to \Spec(\mathcal{O}_{X, x})\) be a modification with \(Y\) normal. We have to show that \(H^1(Y, \mathcal{O}_Y) = 0\). By Limits, Lemma 0B3X we can find a morphism of schemes \(g : X' \to X\) which is an isomorphism over \(X \setminus \{x\}\) such that \(X' \times_X \Spec(\mathcal{O}_{X, x})\) is isomorphic to \(Y\). Then \(g\) is a modification as it is proper by Limits, Lemma 0BFN. The local ring of \(X'\) at a point of \(x'\) is either isomorphic to the local ring of \(X\) at \(g(x')\) if \(g(x') \not = x\) and if \(g(x') = x\), then the local ring of \(X'\) at \(x'\) is isomorphic to the local ring of \(Y\) at the corresponding point. Hence we see that \(X'\) is normal as both \(X\) and \(Y\) are normal. By maximality we have \(R^1g_*\mathcal{O}_{X'} = 0\) (see first paragraph). Clearly this means that \(H^1(Y, \mathcal{O}_Y) = 0\) as desired.

The conclusion is that we’ve found one normal modification \(X\) of \(\Spec(A)\) such that the local rings of \(X\) at closed points all define rational singularities. Then we choose a sequence of normalized blowups \(X_n \to \ldots \to X_1 \to \Spec(A)\) such that \(X_n\) dominates \(X\), see Lemma 0BBT. For a closed point \(x' \in X_n\) mapping to \(x \in X\) we can apply Lemma 0BG0 to the ring map \(\mathcal{O}_{X, x} \to \mathcal{O}_{X_n, x'}\) to see that \(\mathcal{O}_{X_n, x'}\) defines a rational singularity.

Lemma

Let \(A \to B\) be a finite injective local ring map of local normal Nagata domains of dimension \(2\). Assume that the induced extension of fraction fields is separable. If reduction to rational singularities is possible for \(A\) then it is possible for \(B\).

Proof

Let \(n\) be the degree of the fraction field extension \(L/K\). Let \(\text{Trace}_{L/K} : L \to K\) be the trace. Since the extension is finite separable the trace pairing \((h, g) \mapsto \text{Trace}_{L/K}(fg)\) is a nondegenerate bilinear form on \(L\) over \(K\). See Fields, Lemma 0BIL. Pick \(b_1, \ldots, b_n \in B\) which form a basis of \(L\) over \(K\). By the above \(d = \det(\text{Trace}_{L/K}(b_ib_j)) \in A\) is nonzero.

Let \(Y \to \Spec(B)\) be a modification with \(Y\) normal. We can find a \(U\)-admissible blowup \(X'\) of \(\Spec(A)\) such that the strict transform \(Y'\) of \(Y\) is finite over \(X'\), see More on Flatness, Lemma 0B49. Picture \[\xymatrix{ Y' \ar[d] \ar[r] & Y \ar[r] & \Spec(B) \ar[d] \\ X' \ar[rr] & & \Spec(A) }\] After replacing \(X'\) and \(Y'\) by their normalizations we may assume that \(X'\) and \(Y'\) are normal modifications of \(\Spec(A)\) and \(\Spec(B)\). In this way we reduce to the case where there exists a commutative diagram \[\xymatrix{ Y \ar[d]_\pi \ar[r]_-g & \Spec(B) \ar[d] \\ X \ar[r]^-f & \Spec(A) }\] with \(X\) and \(Y\) normal modifications of \(\Spec(A)\) and \(\Spec(B)\) and \(\pi\) finite.

The trace map on \(L\) over \(K\) extends to a map of \(\mathcal{O}_X\)-modules \(\text{Trace} : \pi_*\mathcal{O}_Y \to \mathcal{O}_X\). Consider the map \[\Phi : \pi_*\mathcal{O}_Y \longrightarrow \mathcal{O}_X^{\oplus n},\quad s \longmapsto (\text{Trace}(b_1s), \ldots, \text{Trace}(b_ns))\] This map is injective (because it is injective in the generic point) and there is a map \[\mathcal{O}_X^{\oplus n} \longrightarrow \pi_*\mathcal{O}_Y,\quad (s_1, \ldots, s_n) \longmapsto \sum b_i s_i\] whose composition with \(\Phi\) has matrix \(\text{Trace}(b_ib_j)\). Hence the cokernel of \(\Phi\) is annihilated by \(d\). Thus we see that we have an exact sequence \[H^0(X, \Coker(\Phi)) \to H^1(Y, \mathcal{O}_Y) \to H^1(X, \mathcal{O}_X)^{\oplus n}\] Since the right hand side is bounded by assumption, it suffices to show that the \(d\)-torsion in \(H^1(Y, \mathcal{O}_Y)\) is bounded. This is the content of Lemma 0AXJ and (0AXK).

Lemma

Let \(A\) be a Nagata regular local ring of dimension \(2\). Then \(A\) defines a rational singularity.

Proof

(The assumption that \(A\) be Nagata is not necessary for this proof, but we’ve only defined the notion of rational singularity in the case of Nagata \(2\)-dimensional normal local domains.) Let \(X \to \Spec(A)\) be a modification with \(X\) normal. By Lemma 0AHI we can dominate \(X\) by a scheme \(X_n\) which is the last in a sequence \[X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = \Spec(A)\] of blowing ups in closed points. By Lemma 0AGR the schemes \(X_i\) are regular, in particular normal (Algebra, Lemma 0567). By Lemma 0AXF we have \(H^1(X, \mathcal{O}_X) \subset H^1(X_n, \mathcal{O}_{X_n})\). Thus it suffices to prove \(H^1(X_n, \mathcal{O}_{X_n}) = 0\). Using Lemma 0AXF again, we see that it suffices to prove \(R^1(X_i \to X_{i - 1})_*\mathcal{O}_{X_i} = 0\) for \(i = 1, \ldots, n\). This follows from Lemma 0AGS.

Lemma

Let \(A\) be a local normal Nagata domain of dimension \(2\) which has a dualizing complex \(\omega_A^\bullet\). If there exists a nonzero \(d \in A\) such that for all normal modifications \(X \to \Spec(A)\) the cokernel of the trace map \[\Gamma(X, \omega_X) \to \omega_A\] is annihilated by \(d\), then reduction to rational singularities is possible for \(A\).

Proof

For \(X \to \Spec(A)\) as in the statement we have to bound \(H^1(X, \mathcal{O}_X)\). Let \(\omega_X\) be the dualizing module of \(X\) as in the statement of Grauert-Riemenschneider (Proposition 0AXD). The trace map is the map \(Rf_*\omega_X \to \omega_A\) described in Duality for Schemes, Section 0AWG. By Grauert-Riemenschneider we have \(Rf_*\omega_X = f_*\omega_X\) thus the trace map indeed produces a map \(\Gamma(X, \omega_X) \to \omega_A\). By duality we have \(Rf_*\omega_X = R\Hom_A(Rf_*\mathcal{O}_X, \omega_A)\) (this uses that \(\omega_X[2]\) is the dualizing complex on \(X\) normalized relative to \(\omega_A[2]\), see Duality for Schemes, Lemma 0AUE or more directly Section 0AU3 or even more directly Example 0GEW). The distinguished triangle \[A \to Rf_*\mathcal{O}_X \to R^1f_*\mathcal{O}_X[-1] \to A[1]\] is transformed by \(R\Hom_A(-, \omega_A)\) into the short exact sequence \[0 \to f_*\omega_X \to \omega_A \to \Ext_A^2(R^1f_*\mathcal{O}_X, \omega_A) \to 0\] (and \(\Ext_A^i(R^1f_*\mathcal{O}_X, \omega_A) = 0\) for \(i \not = 2\); this will follow from the discussion below as well). Since \(R^1f_*\mathcal{O}_X\) is supported in \(\{\mathfrak m\}\), the local duality theorem tells us that \[\Ext_A^2(R^1f_*\mathcal{O}_X, \omega_A) = \Ext_A^0(R^1f_*\mathcal{O}_X, \omega_A[2]) = \Hom_A(R^1f_*\mathcal{O}_X, E)\] is the Matlis dual of \(R^1f_*\mathcal{O}_X\) (and the other ext groups are zero), see Dualizing Complexes, Lemma 0AAK. By the equivalence of categories inherent in Matlis duality (Dualizing Complexes, Proposition 08Z9), if \(R^1f_*\mathcal{O}_X\) is not annihilated by \(d\), then neither is the \(\Ext^2\) above. Hence we see that \(H^1(X, \mathcal{O}_X)\) is annihilated by \(d\). Thus the required boundedness follows from Lemma 0AXJ and (0AXK).

Lemma

Let \(p\) be a prime number. Let \(A\) be a regular local ring of dimension \(2\) and characteristic \(p\). Let \(A_0 \subset A\) be a subring such that \(\Omega_{A/A_0}\) is free of rank \(r < \infty\). Set \(\omega_A = \Omega^r_{A/A_0}\). If \(X \to \Spec(A)\) is the result of a sequence of blowups in closed points, then there exists a map \[\varphi_X : (\Omega^r_{X/\Spec(A_0)})^{**} \longrightarrow \omega_X\] extending the given identification in the generic point.

Proof

Observe that \(A\) is Gorenstein (Dualizing Complexes, Lemma 0AWX) and hence the invertible module \(\omega_A\) does indeed serve as a dualizing module. Moreover, any \(X\) as in the lemma has an invertible dualizing module \(\omega_X\) as \(X\) is regular (hence Gorenstein) and proper over \(A\), see Remark 0B4R and Lemma 0AGR. Suppose we have constructed the map \(\varphi_X : (\Omega^r_{X/A_0})^{**} \to \omega_X\) and suppose that \(b : X' \to X\) is a blowup in a closed point. Set \(\Omega^r_X = (\Omega^r_{X/A_0})^{**}\) and \(\Omega^r_{X'} = (\Omega^r_{X'/A_0})^{**}\). Since \(\omega_{X'} = b^!(\omega_X)\) a map \(\Omega^r_{X'} \to \omega_{X'}\) is the same thing as a map \(Rb_*(\Omega^r_{X'}) \to \omega_X\). See discussion in Remark 0B4R and Duality for Schemes, Section 0AU3. Thus in turn it suffices to produce a map \[Rb_*(\Omega^r_{X'}) \longrightarrow \Omega^r_X\] The sheaves \(\Omega^r_{X'}\) and \(\Omega^r_X\) are invertible, see Divisors, Lemma 0B3N. Consider the exact sequence \[b^*\Omega_{X/A_0} \to \Omega_{X'/A_0} \to \Omega_{X'/X} \to 0\] A local calculation shows that \(\Omega_{X'/X}\) is isomorphic to an invertible module on the exceptional divisor \(E\), see Lemma 0B4L. It follows that either \[\Omega^r_{X'} \cong (b^*\Omega^r_X)(E) \quad\text{or}\quad \Omega^r_{X'} \cong b^*\Omega^r_X\] see Divisors, Lemma 0B3T. (The second possibility never happens in characteristic zero, but can happen in characteristic \(p\).) In both cases we see that \(R^1b_*(\Omega^r_{X'}) = 0\) and \(b_*(\Omega^r_{X'}) = \Omega^r_X\) by Lemma 0AGS.

Lemma

Let \(p\) be a prime number. Let \(A\) be a complete regular local ring of dimension \(2\) and characteristic \(p\). Let \(L/K\) be a degree \(p\) inseparable extension of the fraction field \(K\) of \(A\). Let \(B \subset L\) be the integral closure of \(A\). Then reduction to rational singularities is possible for \(B\).

Proof

We have \(A = k[[x, y]]\). Write \(L = K[x]/(x^p - f)\) for some \(f \in A\) and denote \(g \in B\) the congruence class of \(x\), i.e., the element such that \(g^p = f\). By Algebra, Lemma 031W we see that \(\text{d}f\) is nonzero in \(\Omega_{K/\mathbf{F}_p}\). By More on Algebra, Lemma 07P5 there exists a subfield \(k^p \subset k' \subset k\) with \(p^e = [k : k'] < \infty\) such that \(\text{d}f\) is nonzero in \(\Omega_{K/K_0}\) where \(K_0\) is the fraction field of \(A_0 = k'[[x^p, y^p]] \subset A\). Then \[\Omega_{A/A_0} = A \otimes_k \Omega_{k/k'} \oplus A \text{d}x \oplus A \text{d}y\] is finite free of rank \(e + 2\). Set \(\omega_A = \Omega^{e + 2}_{A/A_0}\). Consider the canonical map \[\text{Tr} : \Omega^{e + 2}_{B/A_0} \longrightarrow \Omega^{e + 2}_{A/A_0} = \omega_A\] of Lemma 0AX6. By duality this determines a map \[c : \Omega^{e + 2}_{B/A_0} \to \omega_B = \Hom_A(B, \omega_A)\] Claim: the cokernel of \(c\) is annihilated by a nonzero element of \(B\).

Since \(\text{d}f\) is nonzero in \(\Omega_{A/A_0}\) we can find \(\eta_1, \ldots, \eta_{e + 1} \in \Omega_{A/A_0}\) such that \(\theta = \eta_1 \wedge \ldots \wedge \eta_{e + 1} \wedge \text{d}f\) is nonzero in \(\omega_A = \Omega^{e + 2}_{A/A_0}\). To prove the claim we will construct elements \(\omega_i\) of \(\Omega^{e + 2}_{B/A_0}\), \(i = 0, \ldots, p - 1\) which are mapped to \(\varphi_i \in \omega_B = \Hom_A(B, \omega_A)\) with \(\varphi_i(g^j) = \delta_{ij}\theta\) for \(j = 0, \ldots, p - 1\). Since \(\{1, g, \ldots, g^{p - 1}\}\) is a basis for \(L/K\) this proves the claim. We set \(\eta = \eta_1 \wedge \ldots \wedge \eta_{e + 1}\) so that \(\theta = \eta \wedge \text{d}f\). Set \(\omega_i = \eta \wedge g^{p - 1 - i}\text{d}g\). Then by construction we have \[\varphi_i(g^j) = \text{Tr}(g^j \eta \wedge g^{p - 1 - i}\text{d}g) = \text{Tr}(\eta \wedge g^{p - 1 - i + j}\text{d}g) = \delta_{ij} \theta\] by the explicit description of the trace map in Lemma 0AX5.

Let \(Y \to \Spec(B)\) be a normal modification. Exactly as in the proof of Lemma 0AXL we can reduce to the case where \(Y\) is finite over a modification \(X\) of \(\Spec(A)\). By Lemma 0AHI we may even assume \(X \to \Spec(A)\) is the result of a sequence of blowing ups in closed points. Picture: \[\xymatrix{ Y \ar[d]_\pi \ar[r]_-g & \Spec(B) \ar[d] \\ X \ar[r]^-f & \Spec(A) }\] We may apply Lemma 0AX6 to \(\pi\) and we obtain the first arrow in \[\pi_*(\Omega^{e + 2}_{Y/A_0}) \xrightarrow{\text{Tr}} (\Omega^{e + 2}_{X/A_0})^{**} \xrightarrow{\varphi_X} \omega_X\] and the second arrow is from Lemma 0B4T (because \(f\) is a sequence of blowups in closed points). By duality for the finite morphism \(\pi\) this corresponds to a map \[c_Y : \Omega^{e + 2}_{Y/A_0} \longrightarrow \omega_Y\] extending the map \(c\) above. Hence we see that the image of \(\Gamma(Y, \omega_Y) \to \omega_B\) contains the image of \(c\). By our claim we see that the cokernel is annihilated by a fixed nonzero element of \(B\). We conclude by Lemma 0B4S.

Rational singularities

In this section we reduce from rational singular points to Gorenstein rational singular points. See [Lipman-rational] and [Mattuck].

Situation

Here \((A, \mathfrak m, \kappa)\) be a local normal Nagata domain of dimension \(2\) which defines a rational singularity. Let \(s\) be the closed point of \(S = \Spec(A)\) and \(U = S \setminus \{s\}\). Let \(f : X \to S\) be a modification with \(X\) normal. We denote \(C_1, \ldots, C_r\) the irreducible components of the special fibre \(X_s\) of \(f\).

Lemma

In Situation 0B4W. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Then

1. \(H^p(X, \mathcal{F}) = 0\) for \(p \not \in \{0, 1\}\), and

2. \(H^1(X, \mathcal{F}) = 0\) if \(\mathcal{F}\) is globally generated.

Proof

Part (1) follows from Cohomology of Schemes, Lemma 02V7. If \(\mathcal{F}\) is globally generated, then there is a surjection \(\bigoplus_{i \in I} \mathcal{O}_X \to \mathcal{F}\). By part (1) and the long exact sequence of cohomology this induces a surjection on \(H^1\). Since \(H^1(X, \mathcal{O}_X) = 0\) as \(S\) has a rational singularity, and since \(H^1(X, -)\) commutes with direct sums (Cohomology, Lemma 01FF) we conclude.

Lemma

In Situation 0B4W assume \(E = X_s\) is an effective Cartier divisor. Let \(\mathcal{I}\) be the ideal sheaf of \(E\). Then \(H^0(X, \mathcal{I}^n) = \mathfrak m^n\) and \(H^1(X, \mathcal{I}^n) = 0\).

Proof

We have \(H^0(X, \mathcal{O}_X) = A\), see discussion following Situation 0AX8. Then \(\mathfrak m \subset H^0(X, \mathcal{I}) \subset H^0(X, \mathcal{O}_X)\). The second inclusion is not an equality as \(X_s \not = \emptyset\). Thus \(H^0(X, \mathcal{I}) = \mathfrak m\). As \(\mathcal{I}^n = \mathfrak m^n\mathcal{O}_X\) our Lemma 0B4X shows that \(H^1(X, \mathcal{I}^n) = 0\).

Choose generators \(x_1, \ldots, x_{\mu + 1}\) of \(\mathfrak m\). These define global sections of \(\mathcal{I}\) which generate it. Hence a short exact sequence \[0 \to \mathcal{F} \to \mathcal{O}_X^{\oplus \mu + 1} \to \mathcal{I} \to 0\] Then \(\mathcal{F}\) is a finite locally free \(\mathcal{O}_X\)-module of rank \(\mu\) and \(\mathcal{F} \otimes \mathcal{I}\) is globally generated by Constructions, Lemma 0B3B. Hence \(\mathcal{F} \otimes \mathcal{I}^n\) is globally generated for all \(n \geq 1\). Thus for \(n \geq 2\) we can consider the exact sequence \[0 \to \mathcal{F} \otimes \mathcal{I}^{n - 1} \to (\mathcal{I}^{n - 1})^{\oplus \mu + 1} \to \mathcal{I}^n \to 0\] Applying the long exact sequence of cohomology using that \(H^1(X, \mathcal{F} \otimes \mathcal{I}^{n - 1}) = 0\) by Lemma 0B4X we obtain that every element of \(H^0(X, \mathcal{I}^n)\) is of the form \(\sum x_i a_i\) for some \(a_i \in H^0(X, \mathcal{I}^{n - 1})\). This shows that \(H^0(X, \mathcal{I}^n) = \mathfrak m^n\) by induction.

Lemma

In Situation 0B4W the blowup of \(\Spec(A)\) in \(\mathfrak m\) is normal.

Proof

Let \(X' \to \Spec(A)\) be the blowup, in other words \[X' = \text{Proj}(A \oplus \mathfrak m \oplus \mathfrak m^2 \oplus \ldots).\] is the Proj of the Rees algebra. This in particular shows that \(X'\) is integral and that \(X' \to \Spec(A)\) is a projective modification. Let \(X\) be the normalization of \(X'\). Since \(A\) is Nagata, we see that \(\nu : X \to X'\) is finite (Morphisms, Lemma 035S). Let \(E' \subset X'\) be the exceptional divisor and let \(E \subset X\) be the inverse image. Let \(\mathcal{I}' \subset \mathcal{O}_{X'}\) and \(\mathcal{I} \subset \mathcal{O}_X\) be their ideal sheaves. Recall that \(\mathcal{I}' = \mathcal{O}_{X'}(1)\) (Divisors, Lemma 02NS). Observe that \(\mathcal{I} = \nu^*\mathcal{I}'\) and that \(E\) is an effective Cartier divisor (Divisors, Lemma 02OO). We are trying to show that \(\nu\) is an isomorphism. As \(\nu\) is finite, it suffices to show that \(\mathcal{O}_{X'} \to \nu_*\mathcal{O}_X\) is an isomorphism. If not, then we can find an \(n \geq 0\) such that \[H^0(X', (\mathcal{I}')^n) \not = H^0(X', (\nu_*\mathcal{O}_X) \otimes (\mathcal{I}')^n)\] for example because we can recover quasi-coherent \(\mathcal{O}_{X'}\)-modules from their associated graded modules, see Properties, Lemma 01QJ. By the projection formula we have \[H^0(X', (\nu_*\mathcal{O}_X) \otimes (\mathcal{I}')^n) = H^0(X, \nu^*(\mathcal{I}')^n) = H^0(X, \mathcal{I}^n) = \mathfrak m^n\] the last equality by Lemma 0B4Y. On the other hand, there is clearly an injection \(\mathfrak m^n \to H^0(X', (\mathcal{I}')^n)\). Since \(H^0(X', (\mathcal{I}')^n)\) is torsion free we conclude equality holds for all \(n\), hence \(X = X'\).

Lemma

In Situation 0B4W. Let \(X\) be the blowup of \(\Spec(A)\) in \(\mathfrak m\). Let \(E \subset X\) be the exceptional divisor. With \(\mathcal{O}_X(1) = \mathcal{I}\) as usual and \(\mathcal{O}_E(1) = \mathcal{O}_X(1)|_E\) we have

1. \(E\) is a proper Cohen-Macaulay curve over \(\kappa\).

2. \(\mathcal{O}_E(1)\) is very ample

3. \(\deg(\mathcal{O}_E(1)) \geq 1\) and equality holds only if \(A\) is a regular local ring,

4. \(H^1(E, \mathcal{O}_E(n)) = 0\) for \(n \geq 0\), and

5. \(H^0(E, \mathcal{O}_E(n)) = \mathfrak m^n/\mathfrak m^{n + 1}\) for \(n \geq 0\).

Proof

Since \(\mathcal{O}_X(1)\) is very ample by construction, we see that its restriction to the special fibre \(E\) is very ample as well. By Lemma 0B4Z the scheme \(X\) is normal. Then \(E\) is Cohen-Macaulay by Divisors, Lemma 0B3S. Lemma 0B4Y applies and we obtain (4) and (5) from the exact sequences \[0 \to \mathcal{I}^{n + 1} \to \mathcal{I}^n \to i_*\mathcal{O}_E(n) \to 0\] and the long exact cohomology sequence. In particular, we see that \[\deg(\mathcal{O}_E(1)) = \chi(E, \mathcal{O}_E(1)) - \chi(E, \mathcal{O}_E) = \dim(\mathfrak m/\mathfrak m^2) - 1\] by Varieties, Definition 0AYR. Thus (3) follows as well.

Lemma

In Situation 0B4W assume \(A\) has a dualizing complex \(\omega_A^\bullet\). With \(\omega_X\) the dualizing module of \(X\), the trace map \(H^0(X, \omega_X) \to \omega_A\) is an isomorphism and consequently there is a canonical map \(f^*\omega_A \to \omega_X\).

Proof

By Grauert-Riemenschneider (Proposition 0AXD) we see that \(Rf_*\omega_X = f_*\omega_X\). By duality we have a short exact sequence \[0 \to f_*\omega_X \to \omega_A \to \Ext^2_A(R^1f_*\mathcal{O}_X, \omega_A) \to 0\] (for example see proof of Lemma 0B4S) and since \(A\) defines a rational singularity we obtain \(f_*\omega_X = \omega_A\).

Lemma

In Situation 0B4W assume \(A\) has a dualizing complex \(\omega_A^\bullet\) and is not regular. Let \(X\) be the blowup of \(\Spec(A)\) in \(\mathfrak m\) with exceptional divisor \(E \subset X\). Let \(\omega_X\) be the dualizing module of \(X\). Then

1. \(\omega_E = \omega_X|_E \otimes \mathcal{O}_E(-1)\),

2. \(H^1(X, \omega_X(n)) = 0\) for \(n \geq 0\),

3. the map \(f^*\omega_A \to \omega_X\) of Lemma 0BBU is surjective.

Proof

We will use the results of Lemma 0B63 without further mention. Observe that \(\omega_E = \omega_X|_E \otimes \mathcal{O}_E(-1)\) by Duality for Schemes, Lemmas 0AA4 and 0A9X. Thus \(\omega_X|_E = \omega_E(1)\). Consider the short exact sequences \[0 \to \omega_X(n + 1) \to \omega_X(n) \to i_*\omega_E(n + 1) \to 0\] By Algebraic Curves, Lemma 0B62 we see that \(H^1(E, \omega_E(n + 1)) = 0\) for \(n \geq 0\). Thus we see that the maps \[\ldots \to H^1(X, \omega_X(2)) \to H^1(X, \omega_X(1)) \to H^1(X, \omega_X)\] are surjective. Since \(H^1(X, \omega_X(n))\) is zero for \(n \gg 0\) (Cohomology of Schemes, Lemma 02O1) we conclude that (2) holds.

By Algebraic Curves, Lemma 0B5H we see that \(\omega_X|_E = \omega_E \otimes \mathcal{O}_E(1)\) is globally generated. Since we seen above that \(H^1(X, \omega_X(1)) = 0\) the map \(H^0(X, \omega_X) \to H^0(E, \omega_X|_E)\) is surjective. We conclude that \(\omega_X\) is globally generated hence (3) holds because \(\Gamma(X, \omega_X) = \omega_A\) is used in Lemma 0BBU to define the map.

Lemma

Let \((A, \mathfrak m, \kappa)\) be a local normal Nagata domain of dimension \(2\) which defines a rational singularity. Assume \(A\) has a dualizing complex. Then there exists a finite sequence of blowups in singular closed points \[X = X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = \Spec(A)\] such that \(X_i\) is normal for each \(i\) and such that the dualizing sheaf \(\omega_X\) of \(X\) is an invertible \(\mathcal{O}_X\)-module.

Proof

The dualizing module \(\omega_A\) is a finite \(A\)-module whose stalk at the generic point is invertible. Namely, \(\omega_A \otimes_A K\) is a dualizing module for the fraction field \(K\) of \(A\), hence has rank \(1\). Thus there exists a blowup \(b : Y \to \Spec(A)\) such that the strict transform of \(\omega_A\) with respect to \(b\) is an invertible \(\mathcal{O}_Y\)-module, see Divisors, Lemma 0ESN. By Lemma 0BBT we can choose a sequence of normalized blowups \[X_n \to X_{n - 1} \to \ldots \to X_1 \to \Spec(A)\] such that \(X_n\) dominates \(Y\). By Lemma 0B4Z and arguing by induction each \(X_i \to X_{i - 1}\) is simply a blowing up.

We claim that \(\omega_{X_n}\) is invertible. Since \(\omega_{X_n}\) is a coherent \(\mathcal{O}_{X_n}\)-module, it suffices to see its stalks are invertible modules. If \(x \in X_n\) is a regular point, then this is clear from the fact that regular schemes are Gorenstein (Dualizing Complexes, Lemma 0AWX). If \(x\) is a singular point of \(X_n\), then each of the images \(x_i \in X_i\) of \(x\) is a singular point (because the blowup of a regular point is regular by Lemma 0AGR). Consider the canonical map \(f_n^*\omega_A \to \omega_{X_n}\) of Lemma 0BBU. For each \(i\) the morphism \(X_{i + 1} \to X_i\) is either a blowup of \(x_i\) or an isomorphism at \(x_i\). Since \(x_i\) is always a singular point, it follows from Lemma 0B64 and induction that the maps \(f_i^*\omega_A \to \omega_{X_i}\) is always surjective on stalks at \(x_i\). Hence \[(f_n^*\omega_A)_x \longrightarrow \omega_{X_n, x}\] is surjective. On the other hand, by our choice of \(b\) the quotient of \(f_n^*\omega_A\) by its torsion submodule is an invertible module \(\mathcal{L}\). Moreover, the dualizing module is torsion free (Duality for Schemes, Lemma 0AWK). It follows that \(\mathcal{L}_x \cong \omega_{X_n, x}\) and the proof is complete.

Formal arcs

Let \(X\) be a locally Noetherian scheme. In this section we say that a formal arc in \(X\) is a morphism \(a : T \to X\) where \(T\) is the spectrum of a complete discrete valuation ring \(R\) whose residue field \(\kappa\) is identified with the residue field of the image \(p\) of the closed point of \(\Spec(R)\). Let us say that the formal arc \(a\) is centered at \(p\) in this case. We say the formal arc \(T \to X\) is nonsingular if the induced map \(\mathfrak m_p/\mathfrak m_p^2 \to \mathfrak m_R/\mathfrak m_R^2\) is surjective.

Let \(a : T \to X\), \(T = \Spec(R)\) be a nonsingular formal arc centered at a closed point \(p\) of \(X\). Assume \(X\) is locally Noetherian. Let \(b : X_1 \to X\) be the blowing up of \(X\) at \(x\). Since \(a\) is nonsingular, we see that there is an element \(f \in \mathfrak m_p\) which maps to a uniformizer in \(R\). In particular, we find that the generic point of \(T\) maps to a point of \(X\) not equal to \(p\). In other words, with \(K\) the fraction field of \(R\), the restriction of \(a\) defines a morphism \(\Spec(K) \to X \setminus \{p\}\). Since the morphism \(b\) is proper and an isomorphism over \(X \setminus \{x\}\) we can apply the valuative criterion of properness to obtain a unique morphism \(a_1\) making the following diagram commute \[\xymatrix{ T \ar[r]_{a_1} \ar[rd]_a & X_1 \ar[d]^{b} \\ & X }\] Let \(p_1 \in X_1\) be the image of the closed point of \(T\). Observe that \(p_1\) is a closed point as it is a \(\kappa = \kappa(p)\)-rational point on the fibre of \(X_1 \to X\) over \(x\). Since we have a factorization \[\mathcal{O}_{X, x} \to \mathcal{O}_{X_1, p_1} \to R\] we see that \(a_1\) is a nonsingular formal arc as well.

We can repeat the process and obtain a sequence of blowing ups \[\xymatrix{ T \ar[d]_a \ar[rd]_{a_1} \ar[rrd]_{a_2} \ar[rrrd]^{a_3} \\ (X, p) & (X_1, p_1) \ar[l] & (X_2, p_2) \ar[l] & (X_3, p_3) \ar[l] & \ldots \ar[l] }\] This kind of sequence of blowups can be characterized as follows.

Lemma

Let \(X\) be a locally Noetherian scheme. Let \[(X, p) = (X_0, p_0) \leftarrow (X_1, p_1) \leftarrow (X_2, p_2) \leftarrow (X_3, p_3) \leftarrow \ldots\] be a sequence of blowups such that

1. \(p_i\) is closed, maps to \(p_{i - 1}\), and \(\kappa(p_i) = \kappa(p_{i - 1})\),

2. there exists an \(x_1 \in \mathfrak m_p\) whose image in \(\mathfrak m_{p_i}\), \(i > 0\) defines the exceptional divisor \(E_i \subset X_i\).

Then the sequence is obtained from a nonsingular arc \(a : T \to X\) as above.

Proof

Let us write \(\mathcal{O}_n = \mathcal{O}_{X_n, p_n}\) and \(\mathcal{O} = \mathcal{O}_{X, p}\). Denote \(\mathfrak m \subset \mathcal{O}\) and \(\mathfrak m_n \subset \mathcal{O}_n\) the maximal ideals.

We claim that \(x_1^t \not \in \mathfrak m_n^{t + 1}\). Namely, if this were the case, then in the local ring \(\mathcal{O}_{n + 1}\) the element \(x_1^t\) would be in the ideal of \((t + 1)E_{n + 1}\). This contradicts the assumption that \(x_1\) defines \(E_{n + 1}\).

For every \(n\) choose generators \(y_{n, 1}, \ldots, y_{n, t_n}\) for \(\mathfrak m_n\). As \(\mathfrak m_n \mathcal{O}_{n + 1} = x_1\mathcal{O}_{n + 1}\) by assumption (2), we can write \(y_{n, i} = a_{n, i} x_1\) for some \(a_{n, i} \in \mathcal{O}_{n + 1}\). Since the map \(\mathcal{O}_n \to \mathcal{O}_{n + 1}\) defines an isomorphism on residue fields by (1) we can choose \(c_{n, i} \in \mathcal{O}_n\) having the same residue class as \(a_{n, i}\). Then we see that \[\mathfrak m_n = (x_1, z_{n, 1}, \ldots, z_{n, t_n}), \quad z_{n, i} = y_{n, i} - c_{n, i} x_1\] and the elements \(z_{n, i}\) map to elements of \(\mathfrak m_{n + 1}^2\) in \(\mathcal{O}_{n + 1}\).

Let us consider \[J_n = \Ker(\mathcal{O} \to \mathcal{O}_n/\mathfrak m_n^{n + 1})\] We claim that \(\mathcal{O}/J_n\) has length \(n + 1\) and that \(\mathcal{O}/(x_1) + J_n\) equals the residue field. For \(n = 0\) this is immediate. Assume the statement holds for \(n\). Let \(f \in J_n\). Then in \(\mathcal{O}_n\) we have \[f = a x_1^{n + 1} + x_1^n A_1(z_{n, i}) + x_1^{n - 1} A_2(z_{n, i}) + \ldots + A_{n + 1}(z_{n, i})\] for some \(a \in \mathcal{O}_n\) and some \(A_i\) homogeneous of degree \(i\) with coefficients in \(\mathcal{O}_n\). Since \(\mathcal{O} \to \mathcal{O}_n\) identifies residue fields, we may choose \(a \in \mathcal{O}\) (argue as in the construction of \(z_{n, i}\) above). Taking the image in \(\mathcal{O}_{n + 1}\) we see that \(f\) and \(a x_1^{n + 1}\) have the same image modulo \(\mathfrak m_{n + 1}^{n + 2}\). Since \(x_n^{n + 1} \not \in \mathfrak m_{n + 1}^{n + 2}\) it follows that \(J_n/J_{n + 1}\) has length \(1\) and the claim is true.

Consider \(R = \lim \mathcal{O}/J_n\). This is a quotient of the \(\mathfrak m\)-adic completion of \(\mathcal{O}\) hence it is a complete Noetherian local ring. On the other hand, it is not finite length and \(x_1\) generates the maximal ideal. Thus \(R\) is a complete discrete valuation ring. The map \(\mathcal{O} \to R\) lifts to a local homomorphism \(\mathcal{O}_n \to R\) for every \(n\). There are two ways to show this: (1) for every \(n\) one can use a similar procedure to construct \(\mathcal{O}_n \to R_n\) and then one can show that \(\mathcal{O} \to \mathcal{O}_n \to R_n\) factors through an isomorphism \(R \to R_n\), or (2) one can use Divisors, Lemma 0BFL to show that \(\mathcal{O}_n\) is a localization of a repeated affine blowup algebra to explicitly construct a map \(\mathcal{O}_n \to R\). Having said this it is clear that our sequence of blowups comes from the nonsingular arc \(a : T = \Spec(R) \to X\).

The following lemma is a kind of Néron desingularization lemma.

Lemma

Let \((A, \mathfrak m, \kappa)\) be a Noetherian local domain of dimension \(2\). Let \(A \to R\) be a surjection onto a complete discrete valuation ring. This defines a nonsingular arc \(a : T = \Spec(R) \to \Spec(A)\). Let \[\Spec(A) = X_0 \leftarrow X_1 \leftarrow X_2 \leftarrow X_3 \leftarrow \ldots\] be the sequence of blowing ups constructed from \(a\). If \(A_\mathfrak p\) is a regular local ring where \(\mathfrak p = \Ker(A \to R)\), then for some \(i\) the scheme \(X_i\) is regular at \(x_i\).

Proof

Let \(x_1 \in \mathfrak m\) map to a uniformizer of \(R\). Observe that \(\kappa(\mathfrak p) = K\) is the fraction field of \(R\). Write \(\mathfrak p = (x_2, \ldots, x_r)\) with \(r\) minimal. If \(r = 2\), then \(\mathfrak m = (x_1, x_2)\) and \(A\) is regular and the lemma is true. Assume \(r > 2\). After renumbering if necessary, we may assume that \(x_2\) maps to a uniformizer of \(A_\mathfrak p\). Then \(\mathfrak p/\mathfrak p^2 + (x_2)\) is annihilated by a power of \(x_1\). For \(i > 2\) we can find \(n_i \geq 0\) and \(a_i \in A\) such that \[x_1^{n_i} x_i - a_i x_2 = \sum\nolimits_{2 \leq j \leq k} a_{jk} x_jx_k\] for some \(a_{jk} \in A\). If \(n_i = 0\) for some \(i\), then we can remove \(x_i\) from the list of generators of \(\mathfrak p\) and we win by induction on \(r\). If for some \(i\) the element \(a_i\) is a unit, then we can remove \(x_2\) from the list of generators of \(\mathfrak p\) and we win in the same manner. Thus either \(a_i \in \mathfrak p\) or \(a_i = u_i x_1^{m_1} \bmod \mathfrak p\) for some \(m_1 > 0\) and unit \(u_i \in A\). Thus we have either \[x_1^{n_i} x_i = \sum\nolimits_{2 \leq j \leq k} a_{jk} x_jx_k \quad\text{or}\quad x_1^{n_i} x_i - u_i x_1^{m_i} x_2 = \sum\nolimits_{2 \leq j \leq k} a_{jk} x_jx_k\] We will prove that after blowing up the integers \(n_i\), \(m_i\) decrease which will finish the proof.

Let us see what happens with these equations on the affine blowup algebra \(A' = A[\mathfrak m/x_1]\). As \(\mathfrak m = (x_1, \ldots, x_r)\) we see that \(A'\) is generated over \(R\) by \(y_i = x_i/x_1\) for \(i \geq 2\). Clearly \(A \to R\) extends to \(A' \to R\) with kernel \((y_2, \ldots, y_r)\). Then we see that either \[x_1^{n_i - 1} y_i = \sum\nolimits_{2 \leq j \leq k} a_{jk} y_jy_k \quad\text{or}\quad x_1^{n_i - 1} y_i - u_i x_1^{m_1 - 1} y_2 = \sum\nolimits_{2 \leq j \leq k} a_{jk} y_jy_k\] and the proof is complete.

Base change to the completion

The following simple lemma will turn out to be a useful tool in what follows.

Lemma

Let \((A, \mathfrak m, \kappa)\) be a local ring with finitely generated maximal ideal \(\mathfrak m\). Let \(X\) be a scheme over \(A\). Let \(Y = X \times_{\Spec(A)} \Spec(A^\wedge)\) where \(A^\wedge\) is the \(\mathfrak m\)-adic completion of \(A\). For a point \(q \in Y\) with image \(p \in X\) lying over the closed point of \(\Spec(A)\) the local ring map \(\mathcal{O}_{X, p} \to \mathcal{O}_{Y, q}\) induces an isomorphism on completions.

Proof

We may assume \(X\) is affine. Then we may write \(X = \Spec(B)\). Let \(\mathfrak q \subset B' = B \otimes_A A^\wedge\) be the prime corresponding to \(q\) and let \(\mathfrak p \subset B\) be the prime ideal corresponding to \(p\). By Algebra, Lemma 05GG we have \[B'/(\mathfrak m^\wedge)^n B' = A^\wedge/(\mathfrak m^\wedge)^n \otimes_A B = A/\mathfrak m^n \otimes_A B = B/\mathfrak m^n B\] for all \(n\). Since \(\mathfrak m B \subset \mathfrak p\) and \(\mathfrak m^\wedge B' \subset \mathfrak q\) we see that \(B/\mathfrak p^n\) and \(B'/\mathfrak q^n\) are both quotients of the ring displayed above by the \(n\)th power of the same prime ideal. The lemma follows.

Lemma

Let \((A, \mathfrak m, \kappa)\) be a Noetherian local ring. Let \(X \to \Spec(A)\) be a morphism which is locally of finite type. Set \(Y = X \times_{\Spec(A)} \Spec(A^\wedge)\). Let \(y \in Y\) with image \(x \in X\). Then

1. if \(\mathcal{O}_{Y, y}\) is regular, then \(\mathcal{O}_{X, x}\) is regular,

2. if \(y\) is in the closed fibre, then \(\mathcal{O}_{Y, y}\) is regular \(\Leftrightarrow \mathcal{O}_{X, x}\) is regular, and

3. If \(X\) is proper over \(A\), then \(X\) is regular if and only if \(Y\) is regular.

Proof

Since \(A \to A^\wedge\) is faithfully flat (Algebra, Lemma 00MC), we see that \(Y \to X\) is flat. Hence (1) by Algebra, Lemma 07NG. Lemma 0BG5 shows the morphism \(Y \to X\) induces an isomorphism on complete local rings at points of the special fibres. Thus (2) by More on Algebra, Lemma 07NY. If \(X\) is proper over \(A\), then \(Y\) is proper over \(A^\wedge\) (Morphisms, Lemma 01W4) and we see every closed point of \(X\) and \(Y\) lies in the closed fibre. Thus we see that \(Y\) is a regular scheme if and only if \(X\) is so by Properties, Lemma 02IT.

Lemma

Let \((A, \mathfrak m)\) be a Noetherian local ring with completion \(A^\wedge\). Let \(U \subset \Spec(A)\) and \(U^\wedge \subset \Spec(A^\wedge)\) be the punctured spectra. If \(Y \to \Spec(A^\wedge)\) is a \(U^\wedge\)-admissible blowup, then there exists a \(U\)-admissible blowup \(X \to \Spec(A)\) such that \(Y = X \times_{\Spec(A)} \Spec(A^\wedge)\).

Proof

By definition there exists an ideal \(J \subset A^\wedge\) such that \(V(J) = \{\mathfrak m A^\wedge\}\) and such that \(Y\) is the blowup of \(S^\wedge\) in the closed subscheme defined by \(J\), see Divisors, Definition 080K. Since \(A^\wedge\) is Noetherian this implies \(\mathfrak m^n A^\wedge \subset J\) for some \(n\). Since \(A^\wedge/\mathfrak m^n A^\wedge = A/\mathfrak m^n\) we find an ideal \(\mathfrak m^n \subset I \subset A\) such that \(J = I A^\wedge\). Let \(X \to S\) be the blowup in \(I\). Since \(A \to A^\wedge\) is flat we conclude that the base change of \(X\) is \(Y\) by Divisors, Lemma 0805.

Lemma

Let \((A, \mathfrak m, \kappa)\) be a Nagata local normal domain of dimension \(2\). Assume \(A\) defines a rational singularity and that the completion \(A^\wedge\) of \(A\) is normal. Then

1. \(A^\wedge\) defines a rational singularity, and

2. if \(X \to \Spec(A)\) is the blowing up in \(\mathfrak m\), then for a closed point \(x \in X\) the completion \(\mathcal{O}_{X, x}\) is normal.

Proof

Let \(Y \to \Spec(A^\wedge)\) be a modification with \(Y\) normal. We have to show that \(H^1(Y, \mathcal{O}_Y) = 0\). By Varieties, Lemma 0BFP \(Y \to \Spec(A^\wedge)\) is an isomorphism over the punctured spectrum \(U^\wedge = \Spec(A^\wedge) \setminus \{\mathfrak m^\wedge\}\). By Lemma 0B4M there exists a \(U^\wedge\)-admissible blowup \(Y' \to \Spec(A^\wedge)\) dominating \(Y\). By Lemma 0AFK we find there exists a \(U\)-admissible blowup \(X \to \Spec(A)\) whose base change to \(A^\wedge\) dominates \(Y\). Since \(A\) is Nagata, we can replace \(X\) by its normalization after which \(X \to \Spec(A)\) is a normal modification (but possibly no longer a \(U\)-admissible blowup). Then \(H^1(X, \mathcal{O}_X) = 0\) as \(A\) defines a rational singularity. It follows that \(H^1(X \times_{\Spec(A)} \Spec(A^\wedge), \mathcal{O}_{X \times_{\Spec(A)} \Spec(A^\wedge)}) = 0\) by flat base change (Cohomology of Schemes, Lemma 02KH and flatness of \(A \to A^\wedge\) by Algebra, Lemma 00MB). We find that \(H^1(Y, \mathcal{O}_Y) = 0\) by Lemma 0AXF.

Finally, let \(X \to \Spec(A)\) be the blowing up of \(\Spec(A)\) in \(\mathfrak m\). Then \(Y = X \times_{\Spec(A)} \Spec(A^\wedge)\) is the blowing up of \(\Spec(A^\wedge)\) in \(\mathfrak m^\wedge\). By Lemma 0B4Z we see that both \(Y\) and \(X\) are normal. On the other hand, \(A^\wedge\) is excellent (More on Algebra, Proposition 07QW) hence every affine open in \(Y\) is the spectrum of an excellent normal domain (More on Algebra, Lemma 07QU). Thus for \(y \in Y\) the ring map \(\mathcal{O}_{Y, y} \to \mathcal{O}_{Y, y}^\wedge\) is regular and by More on Algebra, Lemma 0BFK we find that \(\mathcal{O}_{Y, y}^\wedge\) is normal. If \(x \in X\) is a closed point of the special fibre, then there is a unique closed point \(y \in Y\) lying over \(x\). Since \(\mathcal{O}_{X, x} \to \mathcal{O}_{Y, y}\) induces an isomorphism on completions (Lemma 0BG5) we conclude.

Lemma

Let \((A, \mathfrak m)\) be a local Noetherian ring. Let \(X\) be a scheme over \(A\). Assume

1. \(A\) is analytically unramified (Algebra, Definition 032X),

2. \(X\) is locally of finite type over \(A\), and

3. \(X \to \Spec(A)\) is étale at the generic points of irreducible components of \(X\).

Then the normalization of \(X\) is finite over \(X\).

Proof

Since \(A\) is analytically unramified it is reduced by Algebra, Lemma 032Y. Since the normalization of \(X\) depends only on the reduction of \(X\), we may replace \(X\) by its reduction \(X_{red}\); note that \(X_{red} \to X\) is an isomorphism over the open \(U\) where \(X \to \Spec(A)\) is étale because \(U\) is reduced (Descent, Lemma 034E) hence condition (3) remains true after this replacement. In addition we may and do assume that \(X = \Spec(B)\) is affine.

The map \[K = \prod\nolimits_{\mathfrak p \subset A\text{ minimal}} \kappa(\mathfrak p) \longrightarrow K^\wedge = \prod\nolimits_{\mathfrak p^\wedge \subset A^\wedge\text{ minimal}} \kappa(\mathfrak p^\wedge)\] is injective because \(A \to A^\wedge\) is faithfully flat (Algebra, Lemma 00MC) hence induces a surjective map between sets of minimal primes (by going down for flat ring maps, see Algebra, Section 00HU). Both sides are finite products of fields as our rings are Noetherian. Let \(L = \prod_{\mathfrak q \subset B\text{ minimal}} \kappa(\mathfrak q)\). Our assumption (3) implies that \(L = B \otimes_A K\) and that \(K \to L\) is a finite étale ring map (this is true because \(A \to B\) is generically finite, for example use Algebra, Lemma 02ML or the more detailed results in Morphisms, Section 02NV). Since \(B\) is reduced we see that \(B \subset L\). This implies that \[C = B \otimes_A A^\wedge \subset L \otimes_A A^\wedge = L \otimes_K K^\wedge = M\] Then \(M\) is the total ring of fractions of \(C\) and is a finite product of fields as a finite separable algebra over \(K^\wedge\). It follows that \(C\) is reduced and that its normalization \(C'\) is the integral closure of \(C\) in \(M\). The normalization \(B'\) of \(B\) is the integral closure of \(B\) in \(L\). By flatness of \(A \to A^\wedge\) we obtain an injective map \(B' \otimes_A A^\wedge \to M\) whose image is contained in \(C'\). Picture \[B' \otimes_A A^\wedge \longrightarrow C'\] As \(A^\wedge\) is Nagata (by Algebra, Lemma 032W), we see that \(C'\) is finite over \(C = B \otimes_A A^\wedge\) (see Algebra, Lemmas 032W and 03GH). As \(C\) is Noetherian, we conclude that \(B' \otimes_A A^\wedge\) is finite over \(C = B \otimes_A A^\wedge\). Therefore by faithfully flat descent (Algebra, Lemma 03C4) we see that \(B'\) is finite over \(B\) which is what we had to show.

Lemma

Let \((A, \mathfrak m, \kappa)\) be a Noetherian local ring. Let \(X \to \Spec(A)\) be a morphism which is locally of finite type. Set \(Y = X \times_{\Spec(A)} \Spec(A^\wedge)\). If the complement of the special fibre in \(Y\) is normal, then the normalization \(X^\nu \to X\) is finite and the base change of \(X^\nu\) to \(\Spec(A^\wedge)\) recovers the normalization of \(Y\).

Proof

There is an immediate reduction to the case where \(X = \Spec(B)\) is affine with \(B\) a finite type \(A\)-algebra. Set \(C = B \otimes_A A^\wedge\) so that \(Y = \Spec(C)\). Since \(A \to A^\wedge\) is faithfully flat, for any prime \(\mathfrak q \subset B\) there exists a prime \(\mathfrak r \subset C\) lying over \(\mathfrak q\). Then \(B_\mathfrak q \to C_\mathfrak r\) is faithfully flat. Hence if \(\mathfrak q\) does not lie over \(\mathfrak m\), then \(C_\mathfrak r\) is normal by assumption on \(Y\) and we conclude that \(B_\mathfrak q\) is normal by Algebra, Lemma 033G. In this way we see that \(X\) is normal away from the special fibre.

Recall that the complete Noetherian local ring \(A^\wedge\) is Nagata (Algebra, Lemma 032W). Hence the normalization \(Y^\nu \to Y\) is finite (Morphisms, Lemma 035S) and an isomorphism away from the special fibre. Say \(Y^\nu = \Spec(C')\). Then \(C \to C'\) is finite and an isomorphism away from \(V(\mathfrak m C)\). Since \(B \to C\) is flat and induces an isomorphism \(B/\mathfrak m B \to C/\mathfrak m C\) there exists a finite ring map \(B \to B'\) whose base change to \(C\) recovers \(C \to C'\). See More on Algebra, Lemma 0ALK and Remark 05EU. Thus we find a finite morphism \(X' \to X\) which is an isomorphism away from the special fibre and whose base change recovers \(Y^\nu \to Y\). By the discussion in the first paragraph we see that \(X'\) is normal at points not on the special fibre. For a point \(x \in X'\) on the special fibre we have a corresponding point \(y \in Y^\nu\) and a flat map \(\mathcal{O}_{X', x} \to \mathcal{O}_{Y^\nu, y}\). Since \(\mathcal{O}_{Y^\nu, y}\) is normal, so is \(\mathcal{O}_{X', x}\), see Algebra, Lemma 033G. Thus \(X'\) is normal and it follows that it is the normalization of \(X\).

Lemma

Let \((A, \mathfrak m, \kappa)\) be a Noetherian local domain whose completion \(A^\wedge\) is normal. Then given any sequence \[Y_n \to Y_{n - 1} \to \ldots \to Y_1 \to \Spec(A^\wedge)\] of normalized blowups, there exists a sequence of (proper) normalized blowups \[X_n \to X_{n - 1} \to \ldots \to X_1 \to \Spec(A)\] whose base change to \(A^\wedge\) recovers the given sequence.

Proof

Given the sequence \(Y_n \to \ldots \to Y_1 \to Y_0 = \Spec(A^\wedge)\) we inductively construct \(X_n \to \ldots \to X_1 \to X_0 = \Spec(A)\). The base case is \(i = 0\). Given \(X_i\) whose base change is \(Y_i\), let \(Y'_i \to Y_i\) be the blowing up in the closed point \(y_i \in Y_i\) such that \(Y_{i + 1}\) is the normalization of \(Y_i\). Since the closed fibres of \(Y_i\) and \(X_i\) are isomorphic, the point \(y_i\) corresponds to a closed point \(x_i\) on the special fibre of \(X_i\). Let \(X'_i \to X_i\) be the blowup of \(X_i\) in \(x_i\). Then the base change of \(X'_i\) to \(\Spec(A^\wedge)\) is isomorphic to \(Y'_i\). By Lemma 0BG9 the normalization \(X_{i + 1} \to X'_i\) is finite and its base change to \(\Spec(A^\wedge)\) is isomorphic to \(Y_{i + 1}\).

Rational double points

In Section 0B4V we argued that resolution of \(2\)-dimensional rational singularities reduces to the Gorenstein case. A Gorenstein rational surface singularity is a rational double point. We will resolve them by explicit computations.

According to the discussion in Examples, Section 0AL7 there exists a normal Noetherian local domain \(A\) whose completion is isomorphic to \(\mathbf{C}[[x, y, z]]/(z^2)\). In this case one could say that \(A\) has a rational double point singularity, but on the other hand, \(\Spec(A)\) does not have a resolution of singularities. This kind of behaviour cannot occur if \(A\) is a Nagata ring, see Algebra, Lemma 0331.

However, it gets worse as there exists a local normal Nagata domain \(A\) whose completion is \(\mathbf{C}[[x, y, z]]/(yz)\) and another whose completion is \(\mathbf{C}[[x, y, z]]/(y^2 - z^3)\). This is Example 2.5 of [Nishimura-few]. This is why we need to assume the completion of our ring is normal in this section.

Situation

Here \((A, \mathfrak m, \kappa)\) be a Nagata local normal domain of dimension \(2\) which defines a rational singularity, whose completion is normal, and which is Gorenstein. We assume \(A\) is not regular.

The arguments in this section will show that repeatedly blowing up singular points resolves \(\Spec(A)\) in this situation. We will need the following lemma in the course of the proof.

Lemma

Let \(\kappa\) be a field. Let \(I \subset \kappa[x, y]\) be an ideal. Let \[a + b x + c y + d x^2 + exy + f y^2 \in I^2\] for some \(a, b, c, d, e, f \in k\) not all zero. If the colength of \(I\) in \(\kappa[x, y]\) is \(> 1\), then \(a + b x + c y + d x^2 + exy + f y^2 = j(g + hx + iy)^2\) for some \(j, g, h, i \in \kappa\).

Proof

Consider the partial derivatives \(b + 2dx + ey\) and \(c + ex + 2fy\). By the Leibniz rules these are contained in \(I\). If one of these is nonzero, then after a linear change of coordinates, i.e., of the form \(x \mapsto \alpha + \beta x + \gamma y\) and \(y \mapsto \delta + \epsilon x + \zeta y\), we may assume that \(x \in I\). Then we see that \(I = (x)\) or \(I = (x, F)\) with \(F\) a monic polynomial of degree \(\geq 2\) in \(y\). In the first case the statement is clear. In the second case observe that we can write any element in \(I^2\) in the form \[A(x, y) x^2 + B(y) x F + C(y) F^2\] for some \(A(x, y) \in \kappa[x, y]\) and \(B, C \in \kappa[y]\). Thus \[a + b x + c y + d x^2 + exy + f y^2 = A(x, y) x^2 + B(y) x F + C(y) F^2\] and by degree reasons we see that \(B = C = 0\) and \(A\) is a constant.

To finish the proof we need to deal with the case that both partial derivatives are zero. This can only happen in characteristic \(2\) and then we get \[a + d x^2 + f y^2 \in I^2\] We may assume \(f\) is nonzero (if not, then switch the roles of \(x\) and \(y\)). After dividing by \(f\) we obtain the case where the characteristic of \(\kappa\) is \(2\) and \[a + d x^2 + y^2 \in I^2\] If \(a\) and \(d\) are squares in \(\kappa\), then we are done. If not, then there exists a derivation \(\theta : \kappa \to \kappa\) with \(\theta(a) \not = 0\) or \(\theta(d) \not = 0\), see Algebra, Lemma 031W. We can extend this to a derivation of \(\kappa[x, y]\) by setting \(\theta(x) = \theta(y) = 0\). Then we find that \[\theta(a) + \theta(d) x^2 \in I\] The case \(\theta(d) = 0\) is absurd. Thus we may assume that \(\alpha + x^2 \in I\) for some \(\alpha \in \kappa\). Combining with the above we find that \(a + \alpha d + y^2 \in I\). Hence \[J = (\alpha + x^2, a + \alpha d + y^2) \subset I\] with codimension at most \(2\). Observe that \(J/J^2\) is free over \(\kappa[x, y]/J\) with basis \(\alpha + x^2\) and \(a + \alpha d + y^2\). Thus \(a + d x^2 + y^2 = 1 \cdot (a + \alpha d + y^2) + d \cdot (\alpha + x^2) \in I^2\) implies that the inclusion \(J \subset I\) is strict. Thus we find a nonzero element of the form \(g + hx + iy + jxy\) in \(I\). If \(j = 0\), then \(I\) contains a linear form and we can conclude as in the first paragraph. Thus \(j \not = 0\) and \(\dim_\kappa(I/J) = 1\) (otherwise we could find an element as above in \(I\) with \(j = 0\)). We conclude that \(I\) has the form \((\alpha + x^2, \beta + y^2, g + hx + iy + jxy)\) with \(j \not = 0\) and has colength \(3\). In this case \(a + dx^2 + y^2 \in I^2\) is impossible. This can be shown by a direct computation, but we prefer to argue as follows. Namely, to prove this statement we may assume that \(\kappa\) is algebraically closed. Then we can do a coordinate change \(x \mapsto \sqrt{\alpha} + x\) and \(y \mapsto \sqrt{\beta} + y\) and assume that \(I = (x^2, y^2, g' + h'x + i'y + jxy)\) with the same \(j\). Then \(g' = h' = i' = 0\) otherwise the colength of \(I\) is not \(3\). Thus we get \(I = (x^2, y^2, xy)\) and the result is clear.

Let \((A, \mathfrak m, \kappa)\) be as in Situation 0BGC. Let \(X \to \Spec(A)\) be the blowing up of \(\mathfrak m\) in \(\Spec(A)\). By Lemma 0B4Z we see that \(X\) is normal. All singularities of \(X\) are rational singularities by Lemma 0BG0. Since \(\omega_A = A\) we see from Lemma 0B64 that \(\omega_X \cong \mathcal{O}_X\) (see discussion in Remark 0B4R for conventions). Thus all singularities of \(X\) are Gorenstein. Moreover, the local rings of \(X\) at closed point have normal completions by Lemma 0BG7. In other words, by blowing up \(\Spec(A)\) we obtain a normal surface \(X\) whose singular points are as in Situation 0BGC. We will use this below without further mention. (Note: we will see in the course of the discussion below that there are finitely many of these singular points.)

Let \(E \subset X\) be the exceptional divisor. We have \(\omega_E = \mathcal{O}_E(-1)\) by Lemma 0B64. By Lemma 0B63 we have \(\kappa = H^0(E, \mathcal{O}_E)\). Thus \(E\) is a Gorenstein curve and by Riemann-Roch as discussed in Algebraic Curves, Section 0B5B we have \[\chi(E, \mathcal{O}_E) = 1 - g = -(1/2) \deg(\omega_E) = (1/2)\deg(\mathcal{O}_E(1))\] where \(g = \dim_\kappa H^1(E, \mathcal{O}_E) \geq 0\). Since \(\deg(\mathcal{O}_E(1))\) is positive by Varieties, Lemma 0B5Y we find that \(g = 0\) and \(\deg(\mathcal{O}_E(1)) = 2\). It follows that we have \[\dim_\kappa (\mathfrak m^n/\mathfrak m^{n + 1}) = 2n + 1\] by Lemma 0B63 and Riemann-Roch on \(E\).

Choose \(x_1, x_2, x_3 \in \mathfrak m\) which map to a basis of \(\mathfrak m/\mathfrak m^2\). Because \(\dim_\kappa(\mathfrak m^2/\mathfrak m^3) = 5\) the images of \(x_i x_j\), \(i \geq j\) in this \(\kappa\)-vector space satisfy a relation. In other words, we can find \(a_{ij} \in A\), \(i \geq j\), not all contained in \(\mathfrak m\), such that \[a_{11} x_1^2 + a_{12} x_1x_2 + a_{13}x_1x_3 + a_{22} x_2^2 + a_{23} x_2x_3 + a_{33} x_3^2 = \sum a_{ijk} x_ix_jx_k\] for some \(a_{ijk} \in A\) where \(i \leq j \leq k\). Denote \(a \mapsto \overline{a}\) the map \(A \to \kappa\). The quadratic form \(q = \sum \overline{a}_{ij} t_i t_j \in \kappa[t_1, t_2, t_3]\) is well defined up to multiplication by an element of \(\kappa^*\) by our choices. If during the course of our arguments we find that \(\overline{a}_{ij} = 0\) in \(\kappa\), then we can subsume the term \(a_{ij} x_i x_j\) in the right hand side and assume \(a_{ij} = 0\); this operation changes the \(a_{ijk}\) but not the other \(a_{i'j'}\).

The blowing up is covered by \(3\) affine charts corresponding to the “variables” \(x_1, x_2, x_3\). By symmetry it suffices to study one of the charts. To do this let \[A' = A[\mathfrak m/x_1]\] be the affine blowup algebra (as in Algebra, Section 052P). Since \(x_1, x_2, x_3\) generate \(\mathfrak m\) we see that \(A'\) is generated by \(y_2 = x_2/x_1\) and \(y_3 = x_3/x_1\) over \(A\). We will occasionally use \(y_1 = 1\) to simplify formulas. Moreover, looking at our relation above we find that \[a_{11} + a_{12} y_2 + a_{13} y_3 + a_{22} y_2^2 + a_{23} y_2y_3 + a_{33} y_3^2 = x_1 (\sum a_{ijk} y_iy_jy_k)\] in \(A'\). Recall that \(x_1 \in A'\) defines the exceptional divisor \(E\) on our affine open of \(X\) which is therefore scheme theoretically given by \[\kappa[y_2, y_3]/ (\overline{a}_{11} + \overline{a}_{12} y_2 + \overline{a}_{13} y_3 + \overline{a}_{22} y_2^2 + \overline{a}_{23} y_2y_3 + \overline{a}_{33} y_3^2)\] In other words, \(E \subset \mathbf{P}^2_\kappa = \text{Proj}(\kappa[t_1, t_2, t_3])\) is the zero scheme of the quadratic form \(q\) introduced above.

The quadratic form \(q\) is an important invariant of the singularity defined by \(A\). Let us say we are in case II if \(q\) is a square of a linear form times an element of \(\kappa^*\) and in case I otherwise. Observe that we are in case II exactly if, after changing our choice of \(x_1, x_2, x_3\), we have \[x_3^2 = \sum a_{ijk}x_ix_jx_k\] in the local ring \(A\).

Let \(\mathfrak m' \subset A'\) be a maximal ideal lying over \(\mathfrak m\) with residue field \(\kappa'\). In other words, \(\mathfrak m'\) corresponds to a closed point \(p \in E\) of the exceptional divisor. Recall that the surjection \[\kappa[y_2, y_3] \to \kappa'\] has kernel generated by two elements \(f_2, f_3 \in \kappa[y_2, y_3]\) (see for example Algebra, Example 00F0 or the proof of Algebra, Lemma 00OP). Let \(z_2, z_3 \in A'\) map to \(f_2, f_3\) in \(\kappa[y_2, y_3]\). Then we see that \(\mathfrak m' = (x_1, z_2, z_3)\) because \(x_2\) and \(x_3\) become divisible by \(x_1\) in \(A'\).

Claim. If \(X\) is singular at \(p\), then \(\kappa' = \kappa\) or we are in case II. Namely, if \(A'_{\mathfrak m'}\) is singular, then \(\dim_{\kappa'} \mathfrak m'/(\mathfrak m')^2 = 3\) which implies that \(\dim_{\kappa'} \overline{\mathfrak m}'/(\overline{\mathfrak m}')^2 = 2\) where \(\overline{m}'\) is the maximal ideal of \(\mathcal{O}_{E, p} = \mathcal{O}_{X, p}/x_1\mathcal{O}_{X, p}\). This implies that \[q(1, y_2, y_3) = \overline{a}_{11} + \overline{a}_{12} y_2 + \overline{a}_{13} y_3 + \overline{a}_{22} y_2^2 + \overline{a}_{23} y_2y_3 + \overline{a}_{33} y_3^2 \in (f_2, f_3)^2\] otherwise there would be a relation between the classes of \(z_2\) and \(z_3\) in \(\overline{\mathfrak m}'/(\overline{\mathfrak m}')^2\). The claim now follows from Lemma 0BGD.

Resolution in case I. By the claim any singular point of \(X\) is \(\kappa\)-rational. Pick such a singular point \(p\). We may choose our \(x_1, x_2, x_3 \in \mathfrak m\) such that \(p\) lies on the chart described above and has coordinates \(y_2 = y_3 = 0\). Since it is a singular point arguing as in the proof of the claim we find that \(q(1, y_2, y_3) \in (y_2, y_3)^2\). Thus we can choose \(a_{11} = a_{12} = a_{13} = 0\) and \(q(t_1, t_2, t_3) = q(t_2, t_3)\). It follows that \[E = V(q) \subset \mathbf{P}^1_\kappa\] either is the union of two distinct lines meeting at \(p\) or is a degree \(2\) curve with a unique \(\kappa\)-rational point (small detail omitted; use that \(q\) is not a square of a linear form up to a scalar). In both cases we conclude that \(X\) has a unique singular point \(p\) which is \(\kappa\)-rational. We need a bit more information in this case. First, looking at higher terms in the expression above, we find that \(\overline{a}_{111} = 0\) because \(p\) is singular. Then we can write \(a_{111} = b_{111} x_1 \bmod (x_2, x_3)\) for some \(b_{111} \in A\). Then the quadratic form at \(p\) for the generators \(x_1, y_2, y_3\) of \(\mathfrak m'\) is \[q' = \overline{b}_{111} t_1^2 + \overline{a}_{112} t_1 t_2 + \overline{a}_{113} t_1 t_3 + \overline{a}_{22} t_2^2 + \overline{a}_{23} t_2 t_3 + \overline{a}_{33} t_3^2\] We see that \(E' = V(q')\) intersects the line \(t_1 = 0\) in either two points or one point of degree \(2\). We conclude that \(p\) lies in case I.

Suppose that the blowing up \(X' \to X\) of \(X\) at \(p\) again has a singular point \(p'\). Then we see that \(p'\) is a \(\kappa\)-rational point and we can blow up to get \(X'' \to X'\). If this process does not stop we get a sequence of blowings up \[\Spec(A) \leftarrow X \leftarrow X' \leftarrow X'' \leftarrow \ldots\] We want to show that Lemma 0BG2 applies to this situation. To do this we have to say something about the choice of the element \(x_1\) of \(\mathfrak m\). Suppose that \(A\) is in case I and that \(X\) has a singular point. Then we will say that \(x_1 \in \mathfrak m\) is a good coordinate if for any (equivalently some) choice of \(x_2, x_3\) the quadratic form \(q(t_1, t_2, t_3)\) has the property that \(q(0, t_2, t_3)\) is not a scalar times a square. We have seen above that a good coordinate exists. If \(x_1\) is a good coordinate, then the singular point \(p \in E\) of \(X\) does not lie on the hypersurface \(t_1 = 0\) because either this does not have a rational point or if it does, then it is not singular on \(X\). Observe that this is equivalent to the statement that the image of \(x_1\) in \(\mathcal{O}_{X, p}\) cuts out the exceptional divisor \(E\). Now the computations above show that if \(x_1\) is a good coordinate for \(A\), then \(x_1 \in \mathfrak m'\mathcal{O}_{X, p}\) is a good coordinate for \(p\). This of course uses that the notion of good coordinate does not depend on the choice of \(x_2\), \(x_3\) used to do the computation. Hence \(x_1\) maps to a good coordinate at \(p'\), \(p''\), etc. Thus Lemma 0BG2 applies and our sequence of blowing ups comes from a nonsingular arc \(A \to R\). Then the map \(A^\wedge \to R\) is a surjection. Since the completion of \(A\) is normal, we conclude by Lemma 0BG3 that after a finite number of blowups \[\Spec(A^\wedge) \leftarrow X^\wedge \leftarrow (X')^\wedge \leftarrow \ldots\] the resulting scheme \((X^{(n)})^\wedge\) is regular. Since \((X^{(n)})^\wedge \to X^{(n)}\) induces isomorphisms on complete local rings (Lemma 0BG5) we conclude that the same is true for \(X^{(n)}\).

Resolution in case II. Here we have \[x_3^2 = \sum a_{ijk}x_ix_jx_k\] in \(A\) for some choice of generators \(x_1, x_2, x_3\) of \(\mathfrak m\). Then \(q = t_3^2\) and \(E = 2C\) where \(C\) is a line. Recall that in \(A'\) we get \[y_3^2 = x_1(\sum a_{ijk} y_iy_jy_k)\] Since we know that \(X\) is normal, we get a discrete valuation ring \(\mathcal{O}_{X, \xi}\) at the generic point \(\xi\) of \(C\). The element \(y_3 \in A'\) maps to a uniformizer of \(\mathcal{O}_{X, \xi}\). Since \(x_1\) scheme theoretically cuts out \(E\) which is \(C\) with multiplicity \(2\), we see that \(x_1\) is a unit times \(y_3^2\) in \(\mathcal{O}_{X, \xi}\). Looking at our equality above we conclude that \[h(y_2) = \overline{a}_{111} + \overline{a}_{112} y_2 + \overline{a}_{122} y_2^2 + \overline{a}_{222} y_2^3\] must be nonzero in the residue field of \(\xi\). Now, suppose that \(p \in C\) defines a singular point. Then \(y_3\) is zero at \(p\) and \(p\) must correspond to a zero of \(h\) by the reasoning used in proving the claim above. If \(h\) does not have a double zero at \(p\), then the quadratic form \(q'\) at \(p\) is not a square and we conclude that \(p\) falls in case I which we have treated above¹. Since the degree of \(h\) is \(3\) we get at most one singular point \(p \in C\) falling into case II which is moreover \(\kappa\)-rational. After changing our choice of \(x_1, x_2, x_3\) we may assume this is the point \(y_2 = y_3 = 0\). Then \(h = \overline{a}_{122} y_2^2 + \overline{a}_{222} y_2^3\). Moreover, it still has to be the case that \(\overline{a}_{113} = 0\) for the quadratic form \(q'\) to have the right shape. Thus the local ring \(\mathcal{O}_{X, p}\) defines a singularity as in the next paragraph.

The final case we treat is the case where we can choose our generators \(x_1, x_2, x_3\) of \(\mathfrak m\) such that \[x_3^2 + x_1(a x_2^2 + b x_2x_3 + c x_3^2) \in \mathfrak m^4\] for some \(a, b, c \in A\). This is a subclass of case II. If \(\overline{a} = 0\), then we can write \(a = a_1 x_1 + a_2 x_2 + a_3 x_3\) and we get after blowing up \[y_3^2 + x_1(a_1 x_1 y_2^2 + a_2 x_1 y_2^3 + a_3 x_1 y_2^2 y_3 + b y_2 y_3 + c y_3^2) = x_1^2 (\sum a_{ijkl}y_iy_jy_ky_l)\] This means that \(X\) is not normal² a contradiction. By the result of the previous paragraph, if the blowup \(X\) has a singular point \(p\) which falls in case II, then there is only one and it is \(\kappa\)-rational. Computing the affine blowup algebras \(A[\frac{\mathfrak m}{x_2}]\) and \(A[\frac{\mathfrak m}{x_3}]\) the reader easily sees that \(p\) cannot be contained the corresponding opens of \(X\). Thus \(p\) is in the spectrum of \(A[\frac{\mathfrak m}{x_1}]\). Doing the blowing up as before we see that \(p\) must be the point with coordinates \(y_2 = y_3 = 0\) and the new equation looks like \[y_3^2 + x_1(a y_2^2 + b y_2 y_3 + c y_3^2) \in (\mathfrak m')^4\] which has the same shape as before and has the property that \(x_1\) defines the exceptional divisor. Thus if the process does not stop we get an infinite sequence of blowups and on each of these \(x_1\) defines the exceptional divisor in the local ring of the singular point. Thus we can finish the proof using Lemmas 0BG2 and 0BG3 and the same reasoning as before.

Lemma

Let \((A, \mathfrak m, \kappa)\) be a local normal Nagata domain of dimension \(2\) which defines a rational singularity, whose completion is normal, and which is Gorenstein. Then there exists a finite sequence of blowups in singular closed points \[X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = \Spec(A)\] such that \(X_n\) is regular and such that each intervening schemes \(X_i\) is normal with finitely many singular points of the same type.

Proof

This is exactly what was proved in the discussion above.

Implied properties

In this section we prove that for a Noetherian integral scheme the existence of a regular alteration has quite a few consequences. This section should be skipped by those not interested in “bad” Noetherian rings.

Lemma

Let \(Y\) be a Noetherian integral scheme. Assume there exists an alteration \(f : X \to Y\) with \(X\) regular. Then the normalization \(Y^\nu \to Y\) is finite and \(Y\) has a dense open which is regular.

Proof

It suffices to prove this when \(Y = \Spec(A)\) where \(A\) is a Noetherian domain. Let \(B\) be the integral closure of \(A\) in its fraction field. Set \(C = \Gamma(X, \mathcal{O}_X)\). By Cohomology of Schemes, Lemma 02O6 we see that \(C\) is a finite \(A\)-module. As \(X\) is normal (Properties, Lemma 0569) we see that \(C\) is normal domain (Properties, Lemma 0358). Thus \(B \subset C\) and we conclude that \(B\) is finite over \(A\) as \(A\) is Noetherian.

There exists a nonempty open \(V \subset Y\) such that \(f^{-1}V \to V\) is finite, see Morphisms, Definition 0AB0. After shrinking \(V\) we may assume that \(f^{-1}V \to V\) is flat (Morphisms, Proposition 052A). Thus \(f^{-1}V \to V\) is faithfully flat. Then \(V\) is regular by Algebra, Lemma 07NG.

Lemma

Let \((A, \mathfrak m)\) be a local Noetherian ring. Let \(B \subset C\) be finite \(A\)-algebras. Assume that (a) \(B\) is a normal ring, and (b) the \(\mathfrak m\)-adic completion \(C^\wedge\) is a normal ring. Then \(B^\wedge\) is a normal ring.

Proof

Consider the commutative diagram \[\xymatrix{ B \ar[r] \ar[d] & C \ar[d] \\ B^\wedge \ar[r] & C^\wedge }\] Recall that \(\mathfrak m\)-adic completion on the category of finite \(A\)-modules is exact because it is given by tensoring with the flat \(A\)-algebra \(A^\wedge\) (Algebra, Lemma 00MB). We will use Serre’s criterion (Algebra, Lemma 031S) to prove that the Noetherian ring \(B^\wedge\) is normal. Let \(\mathfrak q \subset B^\wedge\) be a prime lying over \(\mathfrak p \subset B\). If \(\dim(B_\mathfrak p) \geq 2\), then \(\text{depth}(B_\mathfrak p) \geq 2\) and since \(B_\mathfrak p \to B^\wedge_\mathfrak q\) is flat we find that \(\text{depth}(B^\wedge_\mathfrak q) \geq 2\) (Algebra, Lemma 0337). If \(\dim(B_\mathfrak p) \leq 1\), then \(B_\mathfrak p\) is either a discrete valuation ring or a field. In that case \(C_\mathfrak p\) is faithfully flat over \(B_\mathfrak p\) (because it is finite and torsion free). Hence \(B^\wedge_\mathfrak p \to C^\wedge_\mathfrak p\) is faithfully flat and the same holds after localizing at \(\mathfrak q\). As \(C^\wedge\) and hence any localization is \((S_2)\) we conclude that \(B^\wedge_\mathfrak p\) is \((S_2)\) by Algebra, Lemma 0352. All in all we find that \((S_2)\) holds for \(B^\wedge\). To prove that \(B^\wedge\) is \((R_1)\) we only have to consider primes \(\mathfrak q \subset B^\wedge\) with \(\dim(B^\wedge_\mathfrak q) \leq 1\). Since \(\dim(B^\wedge_\mathfrak q) = \dim(B_\mathfrak p) + \dim(B^\wedge_\mathfrak q/\mathfrak p B^\wedge_\mathfrak q)\) by Algebra, Lemma 00OM we find that \(\dim(B_\mathfrak p) \leq 1\) and we see that \(B^\wedge_\mathfrak q \to C^\wedge_\mathfrak q\) is faithfully flat as before. We conclude using Algebra, Lemma 0353.

Lemma

Let \((A, \mathfrak m, \kappa)\) be a local Noetherian domain. Assume there exists an alteration \(f : X \to \Spec(A)\) with \(X\) regular. Then

1. there exists a nonzero \(f \in A\) such that \(A_f\) is regular,

2. the integral closure \(B\) of \(A\) in its fraction field is finite over \(A\),

3. the \(\mathfrak m\)-adic completion of \(B\) is a normal ring, i.e., the completions of \(B\) at its maximal ideals are normal domains, and

4. the generic formal fibre of \(A\) is regular.

Proof

Parts (1) and (2) follow from Lemma 0BGG. We have to redo part of the proof of that lemma in order to set up notation for the proof of (3). Set \(C = \Gamma(X, \mathcal{O}_X)\). By Cohomology of Schemes, Lemma 02O6 we see that \(C\) is a finite \(A\)-module. As \(X\) is normal (Properties, Lemma 0569) we see that \(C\) is normal domain (Properties, Lemma 0358). Thus \(B \subset C\) and we conclude that \(B\) is finite over \(A\) as \(A\) is Noetherian. By Lemma 0BGH in order to prove (3) it suffices to show that the \(\mathfrak m\)-adic completion \(C^\wedge\) is normal.

By Algebra, Lemma 07N9 the completion \(C^\wedge\) is the product of the completions of \(C\) at the prime ideals of \(C\) lying over \(\mathfrak m\). There are finitely many of these and these are the maximal ideals \(\mathfrak m_1, \ldots, \mathfrak m_r\) of \(C\). (The corresponding result for \(B\) explains the final statement of the lemma.) Thus replacing \(A\) by \(C_{\mathfrak m_i}\) and \(X\) by \(X_i = X \times_{\Spec(C)} \Spec(C_{\mathfrak m_i})\) we reduce to the case discussed in the next paragraph. (Note that \(\Gamma(X_i, \mathcal{O}) = C_{\mathfrak m_i}\) by Cohomology of Schemes, Lemma 02KH.)

Here \(A\) is a Noetherian local normal domain and \(f : X \to \Spec(A)\) is a regular alteration with \(\Gamma(X, \mathcal{O}_X) = A\). We have to show that the completion \(A^\wedge\) of \(A\) is a normal domain. By Lemma 0BG6 \(Y = X \times_{\Spec(A)} \Spec(A^\wedge)\) is regular. Since \(\Gamma(Y, \mathcal{O}_Y) = A^\wedge\) by Cohomology of Schemes, Lemma 02KH, we conclude that \(A^\wedge\) is normal as before. Namely, \(Y\) is normal by Properties, Lemma 0569. It is connected because \(\Gamma(Y, \mathcal{O}_Y) = A^\wedge\) is local. Hence \(Y\) is normal and integral (as connected and normal implies integral for Noetherian schemes). Thus \(\Gamma(Y, \mathcal{O}_Y) = A^\wedge\) is a normal domain by Properties, Lemma 0358. This proves (3).

Proof of (4). Let \(\eta \in \Spec(A)\) denote the generic point and denote by a subscript \(\eta\) the base change to \(\eta\). Since \(f\) is an alteration, the scheme \(X_\eta\) is finite and faithfully flat over \(\eta\). Since \(Y = X \times_{\Spec(A)} \Spec(A^\wedge)\) is regular by Lemma 0BG6 we see that \(Y_\eta\) is regular (as a limit of opens in \(Y\)). Then \(Y_\eta \to \Spec(A^\wedge \otimes_A \kappa(\eta))\) is finite faithfully flat onto the generic formal fibre. We conclude by Algebra, Lemma 07NG.

Resolution

Here is a definition.

Definition

Let \(Y\) be a Noetherian integral scheme. A resolution of singularities of \(Y\) is a modification \(f : X \to Y\) such that \(X\) is regular.

In the case of surfaces we sometimes want a bit more information.

Definition

Let \(Y\) be a \(2\)-dimensional Noetherian integral scheme. We say \(Y\) has a resolution of singularities by normalized blowups if there exists a sequence \[Y_n \to Y_{n - 1} \to \ldots \to Y_1 \to Y_0 \to Y\] where

1. \(Y_i\) is proper over \(Y\) for \(i = 0, \ldots, n\),

2. \(Y_0 \to Y\) is the normalization,

3. \(Y_i \to Y_{i - 1}\) is a normalized blowup for \(i = 1, \ldots, n\), and

4. \(Y_n\) is regular.

Observe that condition (1) implies that the normalization \(Y_0\) of \(Y\) is finite over \(Y\) and that the normalizations used in the normalized blowing ups are finite as well.

Lemma

Let \((A, \mathfrak m, \kappa)\) be a Noetherian local ring. Assume \(A\) is normal and has dimension \(2\). If \(\Spec(A)\) has a resolution of singularities, then \(\Spec(A)\) has a resolution by normalized blowups.

Proof

By Lemma 0BGI the completion \(A^\wedge\) of \(A\) is normal. By Lemma 0BG6 we see that \(\Spec(A^\wedge)\) has a resolution. By Lemma 0BGA any sequence \(Y_n \to Y_{n - 1} \to \ldots \to \Spec(A^\wedge)\) of normalized blowups of comes from a sequence of normalized blowups \(X_n \to \ldots \to \Spec(A)\). Moreover if \(Y_n\) is regular, then \(X_n\) is regular by Lemma 0BG6. Thus it suffices to prove the lemma in case \(A\) is complete.

Assume in addition \(A\) is a complete. We will use that \(A\) is Nagata (Algebra, Proposition 0335), excellent (More on Algebra, Proposition 07QW), and has a dualizing complex (Dualizing Complexes, Lemma 0BFR). Moreover, the same is true for any ring essentially of finite type over \(A\). If \(B\) is a excellent local normal domain, then the completion \(B^\wedge\) is normal (as \(B \to B^\wedge\) is regular and More on Algebra, Lemma 0BFK applies). We will use this without further mention in the rest of the proof.

Let \(X \to \Spec(A)\) be a resolution of singularities. Choose a sequence of normalized blowing ups \[Y_n \to Y_{n - 1} \to \ldots \to Y_1 \to \Spec(A)\] dominating \(X\) (Lemma 0BBT). The morphism \(Y_n \to X\) is an isomorphism away from finitely many points of \(X\). Hence we can apply Lemma 0AHI to find a sequence of blowing ups \[X_m \to X_{m - 1} \to \ldots \to X\] in closed points such that \(X_m\) dominates \(Y_n\). Diagram \[\xymatrix{ & Y_n \ar[rd] \ar[rr] & & \Spec(A) \\ X_m \ar[rr] \ar[ru] & & X \ar[ru] }\] To prove the lemma it suffices to show that a finite number of normalized blowups of \(Y_n\) produce a regular scheme. By our diagram above we see that \(Y_n\) has a resolution (namely \(X_m\)). As \(Y_n\) is a normal surface this implies that \(Y_n\) has at most finitely many singularities \(y_1, \ldots, y_t\) (because \(X_m \to Y_n\) is an isomorphism away from the fibres of dimension \(1\), see Varieties, Lemma 0BFP).

Let \(x_a \in X\) be the image of \(y_a\). Then \(\mathcal{O}_{X, x_a}\) is regular and hence defines a rational singularity (Lemma 0B4Q). Apply Lemma 0BG0 to \(\mathcal{O}_{X, x_a} \to \mathcal{O}_{Y_n, y_a}\) to see that \(\mathcal{O}_{Y_n, y_a}\) defines a rational singularity. By Lemma 0BBV there exists a finite sequence of blowups in singular closed points \[Y_{a, n_a} \to Y_{a, n_a - 1} \to \ldots \to \Spec(\mathcal{O}_{Y_n, y_a})\] such that \(Y_{a, n_a}\) is Gorenstein, i.e., has an invertible dualizing module. By (the essentially trivial) Lemma 0BFY with \(n' = \sum n_a\) these sequences correspond to a sequence of blowups \[Y_{n + n'} \to Y_{n + n' - 1} \to \ldots \to Y_n\] such that \(Y_{n + n'}\) is normal and the local rings of \(Y_{n + n'}\) are Gorenstein. Using the references given above we can dominate \(Y_{n + n'}\) by a sequence of blowups \(X_{m + m'} \to \ldots \to X_m\) dominating \(Y_{n + n'}\) as in the following \[\xymatrix{ & Y_{n + n'} \ar[rr] & & Y_n \ar[rd] \ar[rr] & & \Spec(A) \\ X_{m + m'} \ar[ru] \ar[rr] & & X_m \ar[rr] \ar[ru] & & X \ar[ru] }\] Thus again \(Y_{n + n'}\) has a finite number of singular points \(y'_1, \ldots, y'_s\), but this time the singularities are rational double points, more precisely, the local rings \(\mathcal{O}_{Y_{n + n'}, y'_b}\) are as in Lemma 0BGE. Arguing exactly as above we conclude that the lemma is true.

Lemma

Let \((A, \mathfrak m, \kappa)\) be a Noetherian complete local ring. Assume \(A\) is a normal domain of dimension \(2\). Then \(\Spec(A)\) has a resolution of singularities.

Proof

A Noetherian complete local ring is J-2 (More on Algebra, Proposition 07PJ), Nagata (Algebra, Proposition 0335), excellent (More on Algebra, Proposition 07QW), and has a dualizing complex (Dualizing Complexes, Lemma 0BFR). Moreover, the same is true for any ring essentially of finite type over \(A\). If \(B\) is a excellent local normal domain, then the completion \(B^\wedge\) is normal (as \(B \to B^\wedge\) is regular and More on Algebra, Lemma 0BFK applies). In other words, the local rings which we encounter in the rest of the proof will have the required “excellency” properties required of them.

Choose \(A_0 \subset A\) with \(A_0\) a regular complete local ring and \(A_0 \to A\) finite, see Algebra, Lemma 032D. This induces a finite extension of fraction fields \(K/K_0\). We will argue by induction on \([K : K_0]\). The base case is when the degree is \(1\) in which case \(A_0 = A\) and the result is true.

Suppose there is an intermediate field \(K_0 \subset L \subset K\), \(K_0 \not = L \not = K\). Let \(B \subset A\) be the integral closure of \(A_0\) in \(L\). By induction we choose a resolution of singularities \(Y \to \Spec(B)\). Let \(X\) be the normalization of \(Y \times_{\Spec(B)} \Spec(A)\). Picture: \[\xymatrix{ X \ar[r] \ar[d] & \Spec(A) \ar[d] \\ Y \ar[r] & \Spec(B) }\] Since \(A\) is J-2 the regular locus of \(X\) is open. Since \(X\) is a normal surface we conclude that \(X\) has at worst finitely many singular points \(x_1, \ldots, x_n\) which are closed points with \(\dim(\mathcal{O}_{X, x_i}) = 2\). For each \(i\) let \(y_i \in Y\) be the image. Since \(\mathcal{O}_{Y, y_i}^\wedge \to \mathcal{O}_{X, x_i}^\wedge\) is finite of smaller degree than before we conclude by induction hypothesis that \(\mathcal{O}_{X, x_i}^\wedge\) has resolution of singularities. By Lemma 0BGM there is a sequence \[Z^\wedge_{i, n_i} \to \ldots \to Z^\wedge_{i, 1} \to \Spec(\mathcal{O}_{X, x_i}^\wedge)\] of normalized blowups with \(Z^\wedge_{i, n_i}\) regular. By Lemma 0BGA there is a corresponding sequence of normalized blowing ups \[Z_{i, n_i} \to \ldots \to Z_{i, 1} \to \Spec(\mathcal{O}_{X, x_i})\] Then \(Z_{i, n_i}\) is a regular scheme by Lemma 0BG6. By Lemma 0BFZ we can fit these normalized blowing ups into a corresponding sequence \[Z_n \to Z_{n - 1} \to \ldots \to Z_1 \to X\] and of course \(Z_n\) is regular too (look at the local rings). This proves the induction step.

Assume there is no intermediate field \(K_0 \subset L \subset K\) with \(K_0 \not = L \not = K\). Then either \(K/K_0\) is separable or the characteristic to \(K\) is \(p\) and \([K : K_0] = p\). Then either Lemma 0AXL or 0B4U implies that reduction to rational singularities is possible. By Lemma 0B4P we conclude that there exists a normal modification \(X \to \Spec(A)\) such that for every singular point \(x\) of \(X\) the local ring \(\mathcal{O}_{X, x}\) defines a rational singularity. Since \(A\) is J-2 we find that \(X\) has finitely many singular points \(x_1, \ldots, x_n\). By Lemma 0BBV there exists a finite sequence of blowups in singular closed points \[X_{i, n_i} \to X_{i, n_i - 1} \to \ldots \to \Spec(\mathcal{O}_{X, x_i})\] such that \(X_{i, n_i}\) is Gorenstein, i.e., has an invertible dualizing module. By (the essentially trivial) Lemma 0BFY with \(n = \sum n_a\) these sequences correspond to a sequence of blowups \[X_n \to X_{n - 1} \to \ldots \to X\] such that \(X_n\) is normal and the local rings of \(X_n\) are Gorenstein. Again \(X_n\) has a finite number of singular points \(x'_1, \ldots, x'_s\), but this time the singularities are rational double points, more precisely, the local rings \(\mathcal{O}_{X_n, x'_i}\) are as in Lemma 0BGE. Arguing exactly as above we conclude that the lemma is true.

We finally come to the main theorem of this chapter.

Theorem

Let \(Y\) be a two dimensional integral Noetherian scheme. The following are equivalent

1. there exists an alteration \(X \to Y\) with \(X\) regular,

2. there exists a resolution of singularities of \(Y\),

3. \(Y\) has a resolution of singularities by normalized blowups,

4. the normalization \(Y^\nu \to Y\) is finite, \(Y^\nu\) has finitely many singular points \(y_1, \ldots, y_m\), and for each \(y_i\) the completion of \(\mathcal{O}_{Y^\nu, y_i}\) is normal.

Proof

The implications (3) \(\Rightarrow\) (2) \(\Rightarrow\) (1) are immediate.

Let \(X \to Y\) be an alteration with \(X\) regular. Then \(Y^\nu \to Y\) is finite by Lemma 0BGG. Consider the factorization \(f : X \to Y^\nu\) from Morphisms, Lemma 035Q. The morphism \(f\) is finite over an open \(V \subset Y^\nu\) containing every point of codimension \(\leq 1\) in \(Y^\nu\) by Varieties, Lemma 0AB7. Then \(f\) is flat over \(V\) by Algebra, Lemma 00R4 and the fact that a normal local ring of dimension \(\leq 2\) is Cohen-Macaulay by Serre’s criterion (Algebra, Lemma 031S). Then \(V\) is regular by Algebra, Lemma 07NG. As \(Y^\nu\) is Noetherian we conclude that \(Y^\nu \setminus V = \{y_1, \ldots, y_m\}\) is finite. By Lemma 0BGI the completion of \(\mathcal{O}_{Y^\nu, y_i}\) is normal. In this way we see that (1) \(\Rightarrow\) (4).

Assume (4). We have to prove (3). We may immediately replace \(Y\) by its normalization. Let \(y_1, \ldots, y_m \in Y\) be the singular points. Applying Lemmas 0BGN and 0BGM we find there exists a finite sequence of normalized blowups \[Y_{i, n_i} \to Y_{i, n_i - 1} \to \ldots \to \Spec(\mathcal{O}^\wedge_{Y, y_i})\] such that \(Y_{i, n_i}\) is regular. By Lemma 0BGA there is a corresponding sequence of normalized blowing ups \[X_{i, n_i} \to \ldots \to X_{i, 1} \to \Spec(\mathcal{O}_{Y, y_i})\] Then \(X_{i, n_i}\) is a regular scheme by Lemma 0BG6. By Lemma 0BFZ we can fit these normalized blowing ups into a corresponding sequence \[X_n \to X_{n - 1} \to \ldots \to X_1 \to Y\] and of course \(X_n\) is regular too (look at the local rings). This completes the proof.

Embedded resolution

Given a curve on a surface there is a blowing up which turns the curve into a strict normal crossings divisor. In this section we will use that a one dimensional locally Noetherian scheme is normal if and only if it is regular (Algebra, Lemma 00PD). We will also use that any point on a locally Noetherian scheme specializes to a closed point (Properties, Lemma 02IL).

Lemma

Let \(Y\) be a one dimensional integral Noetherian scheme. The following are equivalent

1. there exists an alteration \(X \to Y\) with \(X\) regular,

2. there exists a resolution of singularities of \(Y\),

3. there exists a finite sequence \(Y_n \to Y_{n - 1} \to \ldots \to Y_1 \to Y\) of blowups in closed points with \(Y_n\) regular, and

4. the normalization \(Y^\nu \to Y\) is finite.

Proof

The implications (3) \(\Rightarrow\) (2) \(\Rightarrow\) (1) are immediate. The implication (1) \(\Rightarrow\) (4) follows from Lemma 0BGG. Observe that a normal one dimensional scheme is regular hence the implication (4) \(\Rightarrow\) (2) is clear as well. Thus it remains to show that the equivalent conditions (1), (2), and (4) imply (3).

Let \(f : X \to Y\) be a resolution of singularities. Since the dimension of \(Y\) is one we see that \(f\) is finite by Varieties, Lemma 0AB7. We will construct factorizations \[X \to \ldots \to Y_2 \to Y_1 \to Y\] where \(Y_i \to Y_{i - 1}\) is a blowing up of a closed point and not an isomorphism as long as \(Y_{i - 1}\) is not regular. Each of these morphisms will be finite (by the same reason as above) and we will get a corresponding system \[f_*\mathcal{O}_X \supset \ldots \supset f_{2, *}\mathcal{O}_{Y_2} \supset f_{1, *}\mathcal{O}_{Y_1} \supset \mathcal{O}_Y\] where \(f_i : Y_i \to Y\) is the structure morphism. Since \(Y\) is Noetherian, this increasing sequence of coherent submodules must stabilize (Cohomology of Schemes, Lemma 01Y8) which proves that for some \(n\) the scheme \(Y_n\) is regular as desired. To construct \(Y_i\) given \(Y_{i - 1}\) we pick a singular closed point \(y_{i - 1} \in Y_{i - 1}\) and we let \(Y_i \to Y_{i - 1}\) be the corresponding blowup. Since \(X\) is regular of dimension \(1\) (and hence the local rings at closed points are discrete valuation rings and in particular PIDs), the ideal sheaf \(\mathfrak m_{y_{i - 1}} \cdot \mathcal{O}_X\) is invertible. By the universal property of blowing up (Divisors, Lemma 0806) this gives us a factorization \(X \to Y_i\). Finally, \(Y_i \to Y_{i - 1}\) is not an isomorphism as \(\mathfrak m_{y_{i - 1}}\) is not an invertible ideal.

Lemma

Let \(X\) be a Noetherian scheme. Let \(Y \subset X\) be an integral closed subscheme of dimension \(1\) satisfying the equivalent conditions of Lemma 0BI4. Then there exists a finite sequence \[X_n \to X_{n - 1} \to \ldots \to X_1 \to X\] of blowups in closed points such that the strict transform of \(Y\) in \(X_n\) is a regular curve.

Proof

Let \(Y_n \to Y_{n - 1} \to \ldots \to Y_1 \to Y\) be the sequence of blowups given to us by Lemma 0BI4. Let \(X_n \to X_{n - 1} \to \ldots \to X_1 \to X\) be the corresponding sequence of blowups of \(X\). This works because the strict transform is the blowup by Divisors, Lemma 080E.

Let \(X\) be a locally Noetherian scheme. Let \(Y, Z \subset X\) be closed subschemes. Let \(p \in Y \cap Z\) be a closed point. Assume that \(Y\) is integral of dimension \(1\) and that the generic point of \(Y\) is not contained in \(Z\). In this situation we can consider the invariant [0BI6]\[\begin{equation} m_p(Y \cap Z) = \text{length}_{\mathcal{O}_{X, p}}(\mathcal{O}_{Y \cap Z, p}) \end{equation}\] This is an integer \(\geq 1\). Namely, if \(I, J \subset \mathcal{O}_{X, p}\) are the ideals corresponding to \(Y, Z\), then we see that \(\mathcal{O}_{Y \cap Z, p} = \mathcal{O}_{X, p}/I + J\) has support equal to \(\{\mathfrak m_p\}\) because we assumed that \(Y \cap Z\) does not contain the unique point of \(Y\) specializing to \(p\). Hence the length is finite by Algebra, Lemma 00L5.

Lemma

In the situation above let \(X' \to X\) be the blowing up of \(X\) in \(p\). Let \(Y', Z' \subset X'\) be the strict transforms of \(Y, Z\). If \(\mathcal{O}_{Y, p}\) is regular, then

1. \(Y' \to Y\) is an isomorphism,

2. \(Y'\) meets the exceptional fibre \(E \subset X'\) in one point \(q\) and \(m_q(Y \cap E) = 1\),

3. if \(q \in Z'\) too, then \(m_q(Y \cap Z') < m_p(Y \cap Z)\).

Proof

Since \(\mathcal{O}_{X, p} \to \mathcal{O}_{Y, p}\) is surjective and \(\mathcal{O}_{Y, p}\) is a discrete valuation ring, we can pick an element \(x_1 \in \mathfrak m_p\) mapping to a uniformizer in \(\mathcal{O}_{Y, p}\). Choose an affine open \(U = \Spec(A)\) containing \(p\) such that \(x_1 \in A\). Let \(\mathfrak m \subset A\) be the maximal ideal corresponding to \(p\). Let \(I, J \subset A\) be the ideals defining \(Y, Z\) in \(\Spec(A)\). After shrinking \(U\) we may assume that \(\mathfrak m = I + (x_1)\), in other words, that \(V(x_1) \cap U \cap Y = \{p\}\) scheme theoretically. We conclude that \(p\) is an effective Cartier divisor on \(Y\) and since \(Y'\) is the blowing up of \(Y\) in \(p\) (Divisors, Lemma 080E) we see that \(Y' \to Y\) is an isomorphism by Divisors, Lemma 0807. The relationship \(\mathfrak m = I + (x_1)\) implies that \(\mathfrak m^n \subset I + (x_1^n)\) hence we can define a map \[\psi : A[\textstyle{\frac{\mathfrak m}{x_1}}] \longrightarrow A/I\] by sending \(y/x_1^n \in A[\frac{\mathfrak m}{x_1}]\) to the class of \(a\) in \(A/I\) where \(a\) is chosen such that \(y \equiv ax_1^n \bmod I\). Then \(\psi\) corresponds to the morphism of \(Y \cap U\) into \(X'\) over \(U\) given by \(Y' \cong Y\). Since the image of \(x_1\) in \(A[\frac{\mathfrak m}{x_1}]\) cuts out the exceptional divisor we conclude that \(m_q(Y', E) = 1\). Finally, since \(J \subset \mathfrak m\) implies that the ideal \(J' \subset A[\frac{\mathfrak m}{x_1}]\) certainly contains the elements \(f/x_1\) for \(f \in J\). Thus if we choose \(f \in J\) whose image \(\overline{f}\) in \(A/I\) has minimal valuation equal to \(m_p(Y \cap Z)\), then we see that \(\psi(f/x_1) = \overline{f}/x_1\) in \(A/I\) has valuation one less proving the last part of the lemma.

Lemma

Let \(X\) be a Noetherian scheme. Let \(Y_i \subset X\), \(i = 1, \ldots, n\) be an integral closed subschemes of dimension \(1\) each satisfying the equivalent conditions of Lemma 0BI4. Then there exists a finite sequence \[X_n \to X_{n - 1} \to \ldots \to X_1 \to X\] of blowups in closed points such that the strict transform \(Y'_i \subset X_n\) of \(Y_i\) in \(X_n\) are pairwise disjoint regular curves.

Proof

It follows from Lemma 0BI5 that we may assume \(Y_i\) is a regular curve for \(i = 1, \ldots, n\). For every \(i \not = j\) and \(p \in Y_i \cap Y_j\) we have the invariant \(m_p(Y_i \cap Y_j)\) (0BI6). If the maximum of these numbers is \(> 1\), then we can decrease it (Lemma 0BI7) by blowing up in all the points \(p\) where the maximum is attained. If the maximum is \(1\) then we can separate the curves using the same lemma by blowing up in all these points \(p\).

When our curve is contained on a regular surface we often want to turn it into a divisor with normal crossings.

Lemma

Let \(X\) be a regular scheme of dimension \(2\). Let \(Z \subset X\) be a proper closed subscheme. There exists a sequence \[X_n \to \ldots \to X_1 \to X\] of blowing ups in closed points such that the inverse image \(Z_n\) of \(Z\) in \(X_n\) is an effective Cartier divisor.

Proof

Let \(D \subset Z\) be the largest effective Cartier divisor contained in \(Z\). Then \(\mathcal{I}_Z \subset \mathcal{I}_D\) and the quotient is supported in closed points by Divisors, Lemma 0AGB. Thus we can write \(\mathcal{I}_Z = \mathcal{I}_{Z'} \mathcal{I}_D\) where \(Z' \subset X\) is a closed subscheme which set theoretically consists of finitely many closed points. Applying Lemma 0AHH we find a sequence of blowups as in the statement of our lemma such that \(\mathcal{I}_{Z'}\mathcal{O}_{X_n}\) is invertible. This proves the lemma.

Lemma

Let \(X\) be a regular scheme of dimension \(2\). Let \(Z \subset X\) be a proper closed subscheme such that every irreducible component \(Y \subset Z\) of dimension \(1\) satisfies the equivalent conditions of Lemma 0BI4. Then there exists a sequence \[X_n \to \ldots \to X_1 \to X\] of blowups in closed points such that the inverse image \(Z_n\) of \(Z\) in \(X_n\) is an effective Cartier divisor supported on a strict normal crossings divisor.

Proof

Let \(X' \to X\) be a blowup in a closed point \(p\). Then the inverse image \(Z' \subset X'\) of \(Z\) is supported on the strict transform of \(Z\) and the exceptional divisor. The exceptional divisor is a regular curve (Lemma 0AGQ) and the strict transform \(Y'\) of each irreducible component \(Y\) is either equal to \(Y\) or the blowup of \(Y\) at \(p\). Thus in this process we do not produce additional singular components of dimension \(1\). Thus it follows from Lemmas 0BIB and 0BI8 that we may assume \(Z\) is an effective Cartier divisor and that all irreducible components \(Y\) of \(Z\) are regular. (Of course we cannot assume the irreducible components are pairwise disjoint because in each blowup of a point of \(Z\) we add a new irreducible component to \(Z\), namely the exceptional divisor.)

Assume \(Z\) is an effective Cartier divisor whose irreducible components \(Y_i\) are regular. For every \(i \not = j\) and \(p \in Y_i \cap Y_j\) we have the invariant \(m_p(Y_i \cap Y_j)\) (0BI6). If the maximum of these numbers is \(> 1\), then we can decrease it (Lemma 0BI7) by blowing up in all the points \(p\) where the maximum is attained (note that the “new” invariants \(m_{q_i}(Y'_i \cap E)\) are always \(1\)). If the maximum is \(1\) then, if \(p \in Y_1 \cap \ldots \cap Y_r\) for some \(r > 2\) and not any of the others (for example), then after blowing up \(p\) we see that \(Y'_1, \ldots, Y'_r\) do not meet in points above \(p\) and \(m_{q_i}(Y'_i, E) = 1\) where \(Y'_i \cap E = \{q_i\}\). Thus continuing to blowup points where more than \(3\) of the components of \(Z\) meet, we reach the situation where for every closed point \(p \in X\) there is either (a) no curves \(Y_i\) passing through \(p\), (b) exactly one curve \(Y_i\) passing through \(p\) and \(\mathcal{O}_{Y_i, p}\) is regular, or (c) exactly two curves \(Y_i\), \(Y_j\) passing through \(p\), the local rings \(\mathcal{O}_{Y_i, p}\), \(\mathcal{O}_{Y_j, p}\) are regular and \(m_p(Y_i \cap Y_j) = 1\). This means that \(\sum Y_i\) is a strict normal crossings divisor on the regular surface \(X\), see Étale Morphisms, Lemma 0BIA.

Contracting exceptional curves

Let \(X\) be a Noetherian scheme. Let \(E \subset X\) be a closed subscheme with the following properties

1. \(E\) is an effective Cartier divisor on \(X\),

2. there exists a field \(k\) and an isomorphism \(\mathbf{P}^1_k \to E\) of schemes,

3. the normal sheaf \(\mathcal{N}_{E/X}\) pulls back to \(\mathcal{O}_{\mathbf{P}^1}(-1)\).

Such a closed subscheme is called an exceptional curve of the first kind.

Let \(X'\) be a Noetherian scheme and let \(x \in X'\) be a closed point such that \(\mathcal{O}_{X', x}\) is regular of dimension \(2\). Let \(b : X \to X'\) be the blowing up of \(X'\) at \(x\). In this case the exceptional fibre \(E \subset X\) is an exceptional curve of the first kind. This follows from Lemma 0AGQ.

Question: Is every exceptional curve of the first kind obtained as the fibre of a blowing up as above? In other words, does there always exist a proper morphism of schemes \(X \to X'\) such that \(E\) maps to a closed point \(x \in X'\), such that \(\mathcal{O}_{X', x}\) is regular of dimension \(2\), and such that \(X\) is the blowing up of \(X'\) at \(x\). If true we say there exists a contraction of \(E\).

Lemma

Let \(X\) be a Noetherian scheme. Let \(E \subset X\) be an exceptional curve of the first kind. If a contraction \(X \to X'\) of \(E\) exists, then it has the following universal property: for every morphism \(\varphi : X \to Y\) such that \(\varphi(E)\) is a point, there is a unique factorization \(X \to X' \to Y\) of \(\varphi\).

Proof

Let \(b : X \to X'\) be a contraction of \(E\). As a topological space \(X'\) is the quotient of \(X\) by the relation identifying all points of \(E\) to one point. Namely, \(b\) is proper (Divisors, Lemma 02NS and Morphisms, Lemma 01WC) and surjective, hence defines a submersive map of topological spaces (Topology, Lemma 0AAU). On the other hand, the canonical map \(\mathcal{O}_{X'} \to b_*\mathcal{O}_X\) is an isomorphism. Namely, this is clear over the complement of the image point \(x \in X'\) of \(E\) and on stalks at \(x\) the map is an isomorphism by part (4) of Lemma 0AGS. Thus the pair \((X', \mathcal{O}_{X'})\) is constructed from \(X\) by taking the quotient as a topological space and endowing this with \(b_*\mathcal{O}_X\) as structure sheaf.

Given \(\varphi\) we can let \(\varphi' : X' \to Y\) be the unique map of topological spaces such that \(\varphi = \varphi' \circ b\). Then the map \[\varphi^\sharp : \varphi^{-1}\mathcal{O}_Y = b^{-1}((\varphi')^{-1}\mathcal{O}_Y) \to \mathcal{O}_X\] is adjoint to a map \[(\varphi')^\sharp : (\varphi')^{-1}\mathcal{O}_Y \to b_*\mathcal{O}_X = \mathcal{O}_{X'}\] Then \((\varphi', (\varphi')^\sharp)\) is a morphism of ringed spaces from \(X'\) to \(Y\) such that we get the desired factorization. Since \(\varphi\) is a morphism of locally ringed spaces, it follows that \(\varphi'\) is too. Namely, the only thing to check is that the map \(\mathcal{O}_{Y, y} \to \mathcal{O}_{X', x}\) is local, where \(y \in Y\) is the image of \(E\) under \(\varphi\). This is true because an element \(f \in \mathfrak m_y\) pulls back to a function on \(X\) which is zero in every point of \(E\) hence the pull back of \(f\) to \(X'\) is a function defined on a neighbourhood of \(x\) in \(X'\) with the same property. Then it is clear that this function must vanish at \(x\) as desired.

Lemma

Let \(X\) be a Noetherian scheme. Let \(E \subset X\) be an exceptional curve of the first kind. If there exists a contraction of \(E\), then it is unique up to unique isomorphism.

Proof

This is immediate from the universal property of Lemma 0C5J.

Lemma

Let \(X\) be a Noetherian scheme. Let \(E \subset X\) be an exceptional curve of the first kind. Let \(E_n = nE\) and denote \(\mathcal{O}_n\) its structure sheaf. Then \[A = \lim H^0(E_n, \mathcal{O}_n)\] is a complete local Noetherian regular local ring of dimension \(2\) and \(\Ker(A \to H^0(E_n, \mathcal{O}_n))\) is the \(n\)th power of its maximal ideal.

Proof

Recall that there exists an isomorphism \(\mathbf{P}^1_k \to E\) such that the normal sheaf of \(E\) in \(X\) pulls back to \(\mathcal{O}(-1)\). Then \(H^0(E, \mathcal{O}_E) = k\). We will denote \(\mathcal{O}_n(iE)\) the restriction of the invertible sheaf \(\mathcal{O}_X(iE)\) to \(E_n\) for all \(n \geq 1\) and \(i \in \mathbf{Z}\). Recall that \(\mathcal{O}_X(-nE)\) is the ideal sheaf of \(E_n\). Hence for \(d \geq 0\) we obtain a short exact sequence \[0 \to \mathcal{O}_E(-(d + n)E) \to \mathcal{O}_{n + 1}(-dE) \to \mathcal{O}_n(-dE) \to 0\] Since \(\mathcal{O}_E(-(d + n)E) = \mathcal{O}_{\mathbf{P}^1_k}(d + n)\) the first cohomology group vanishes for all \(d \geq 0\) and \(n \geq 1\). We conclude that the transition maps of the system \(H^0(E_n, \mathcal{O}_n(-dE))\) are surjective. For \(d = 0\) we get an inverse system of surjections of rings such that the kernel of each transition map is a nilpotent ideal. Hence \(A = \lim H^0(E_n, \mathcal{O}_n)\) is a local ring with residue field \(k\) and maximal ideal \[\lim \Ker(H^0(E_n, \mathcal{O}_n) \to H^0(E, \mathcal{O}_E)) = \lim H^0(E_n, \mathcal{O}_n(-E))\] Pick \(x, y\) in this kernel mapping to a \(k\)-basis of \(H^0(E, \mathcal{O}_E(-E)) = H^0(\mathbf{P}^1_k, \mathcal{O}(1))\). Then \(x^d, x^{d - 1}y, \ldots, y^d\) are elements of \(\lim H^0(E_n, \mathcal{O}_n(-dE))\) which map to a basis of \(H^0(E, \mathcal{O}_E(-dE)) = H^0(\mathbf{P}^1_k, \mathcal{O}(d))\). In this way we see that \(A\) is separated and complete with respect to the linear topology defined by the kernels \[I_n = \Ker(A \longrightarrow H^0(E_n, \mathcal{O}_n))\] We have \(x, y \in I_1\), \(I_d I_{d'} \subset I_{d + d'}\) and \(I_d/I_{d + 1}\) is a free \(k\)-module on \(x^d, x^{d - 1}y, \ldots, y^d\). We will show that \(I_d = (x, y)^d\). Namely, if \(z_e \in I_e\) with \(e \geq d\), then we can write \[z_e = a_{e, 0} x^d + a_{e, 1} x^{d - 1}y + \ldots + a_{e, d}y^d + z_{e + 1}\] where \(a_{e, j} \in (x, y)^{e - d}\) and \(z_{e + 1} \in I_{e + 1}\) by our description of \(I_d/I_{d + 1}\). Thus starting with some \(z = z_d \in I_d\) we can do this inductively \[z = \sum\nolimits_{e \geq d} \sum\nolimits_j a_{e, j} x^{d - j} y^j\] with some \(a_{e, j} \in (x, y)^{e - d}\). Then \(a_j = \sum_{e \geq d} a_{e, j}\) exists (by completeness and the fact that \(a_{e, j} \in I_{e - d}\)) and we have \(z = \sum a_{e, j} x^{d - j} y^j\). Hence \(I_d = (x, y)^d\). Thus \(A\) is \((x, y)\)-adically complete. Then \(A\) is Noetherian by Algebra, Lemma 05GH. It is clear that the dimension is \(2\) by the description of \((x, y)^d/(x, y)^{d + 1}\) and Algebra, Proposition 00KQ. Since the maximal ideal is generated by two elements it is regular.

Lemma

Let \(X\) be a Noetherian scheme. Let \(E \subset X\) be an exceptional curve of the first kind. If there exists a morphism \(f : X \to Y\) such that

1. \(Y\) is Noetherian,

2. \(f\) is proper,

3. \(f\) maps \(E\) to a point \(y\) of \(Y\),

4. \(f\) is quasi-finite at every point not in \(E\),

Then there exists a contraction of \(E\) and it is the Stein factorization of \(f\).

Proof

We apply More on Morphisms, Theorem 03H0 to get a Stein factorization \(X \to X' \to Y\). Then \(X \to X'\) satisfies all the hypotheses of the lemma (some details omitted). Thus after replacing \(Y\) by \(X'\) we may in addition assume that \(f_*\mathcal{O}_X = \mathcal{O}_Y\) and that the fibres of \(f\) are geometrically connected.

Assume that \(f_*\mathcal{O}_X = \mathcal{O}_Y\) and that the fibres of \(f\) are geometrically connected. Note that \(y \in Y\) is a closed point as \(f\) is closed and \(E\) is closed. The restriction \(f^{-1}(Y \setminus \{y\}) \to Y \setminus \{y\}\) of \(f\) is a finite morphism (More on Morphisms, Lemma 02LS). Hence this restriction is an isomorphism since \(f_*\mathcal{O}_X = \mathcal{O}_Y\) since finite morphisms are affine. To prove that \(\mathcal{O}_{Y, y}\) is regular of dimension \(2\) we consider the isomorphism \[\mathcal{O}_{Y, y}^\wedge \longrightarrow \lim H^0(X \times_Y \Spec(\mathcal{O}_{Y, y}/\mathfrak m_y^n), \mathcal{O})\] of Cohomology of Schemes, Lemma 02OD. Let \(E_n = nE\) as in Lemma 0C2K. Observe that \[E_n \subset X \times_Y \Spec(\mathcal{O}_{Y, y}/\mathfrak m_y^n)\] because \(E \subset X_y = X \times_Y \Spec(\kappa(y))\). On the other hand, since \(E = f^{-1}(\{y\})\) set theoretically (because the fibres of \(f\) are geometrically connected), we see that the scheme theoretic fibre \(X_y\) is scheme theoretically contained in \(E_n\) for some \(n > 0\). Namely, apply Cohomology of Schemes, Lemma 01Y9 to the coherent \(\mathcal{O}_X\)-module \(\mathcal{F} = \mathcal{O}_{X_y}\) and the ideal sheaf \(\mathcal{I}\) of \(E\) and use that \(\mathcal{I}^n\) is the ideal sheaf of \(E_n\). This shows that \[X \times_Y \Spec(\mathcal{O}_{Y, y}/\mathfrak m_y^m) \subset E_{nm}\] Thus the inverse limit displayed above is equal to \(\lim H^0(E_n, \mathcal{O}_n)\) which is a regular two dimensional local ring by Lemma 0C2K. Hence \(\mathcal{O}_{Y, y}\) is a two dimensional regular local ring because its completion is so (More on Algebra, Lemma 07NY and 07NV).

We still have to prove that \(f : X \to Y\) is the blowup \(b : Y' \to Y\) of \(Y\) at \(y\). We encourage the reader to find her own proof. First, we note that Lemma 0C2K also implies that \(X_y = E\) scheme theoretically. Since the ideal sheaf of \(E\) is invertible, this shows that \(f^{-1}\mathfrak m_y \cdot \mathcal{O}_X\) is invertible. Hence we obtain a factorization \[X \to Y' \to Y\] of the morphism \(f\) by the universal property of blowing up, see Divisors, Lemma 0806. Recall that the exceptional fibre of \(E' \subset Y'\) is an exceptional curve of the first kind by Lemma 0AGQ. Let \(g : E \to E'\) be the induced morphism. Because for both \(E'\) and \(E\) the conormal sheaf is generated by (pullbacks of) \(a\) and \(b\), we see that the canonical map \(g^*\mathcal{C}_{E'/Y'} \to \mathcal{C}_{E/X}\) (Morphisms, Lemma 01R4) is surjective. Since both are invertible, this map is an isomorphism. Since \(\mathcal{C}_{E/X}\) has positive degree, it follows that \(g\) cannot be a constant morphism. Hence \(g\) has finite fibres. Hence \(g\) is a finite morphism (same reference as above). However, since \(Y'\) is regular (and hence normal) at all points of \(E'\) and since \(X \to Y'\) is birational and an isomorphism away from \(E'\), we conclude that \(X \to Y'\) is an isomorphism by Varieties, Lemma 0BFP.

Lemma

Let \(b : X \to X'\) be the contraction of an exceptional curve of the first kind \(E \subset X\). Then there is a short exact sequence \[0 \to \Pic(X') \to \Pic(X) \to \mathbf{Z} \to 0\] where the first map is pullback by \(b\) and the second map sends \(\mathcal{L}\) to the degree of \(\mathcal{L}\) on the exceptional curve \(E\). The sequence is split by the map \(n \mapsto \mathcal{O}_X(-nE)\).

Proof

Since \(E = \mathbf{P}^1_k\) we see that the Picard group of \(E\) is \(\mathbf{Z}\), see Divisors, Lemma 0BXJ. Hence we can think of the last map as \(\mathcal{L} \mapsto \mathcal{L}|_E\). The degree of the restriction of \(\mathcal{O}_X(E)\) to \(E\) is \(-1\) by definition of exceptional curves of the first kind. Combining these remarks we see that it suffices to show that \(\Pic(X') \to \Pic(X)\) is injective with image the invertible sheaves restricting to \(\mathcal{O}_E\) on \(E\).

Given an invertible \(\mathcal{O}_{X'}\)-module \(\mathcal{L}'\) we claim the map \(\mathcal{L}' \to b_*b^*\mathcal{L}'\) is an isomorphism. This is clear everywhere except possibly at the image point \(x \in X'\) of \(E\). To check it is an isomorphism on stalks at \(x\) we may replace \(X'\) by an open neighbourhood at \(x\) and assume \(\mathcal{L}'\) is \(\mathcal{O}_{X'}\). Then we have to show that the map \(\mathcal{O}_{X'} \to b_*\mathcal{O}_X\) is an isomorphism. This follows from Lemma 0AGS part (4).

Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module with \(\mathcal{L}|_E = \mathcal{O}_E\). Then we claim (1) \(b_*\mathcal{L}\) is invertible and (2) \(b^*b_*\mathcal{L} \to \mathcal{L}\) is an isomorphism. Statements (1) and (2) are clear over \(X' \setminus \{x\}\). Thus it suffices to prove (1) and (2) after base change to \(\Spec(\mathcal{O}_{X', x})\). Computing \(b_*\) commutes with flat base change (Cohomology of Schemes, Lemma 02KH) and similarly for \(b^*\) and formation of the adjunction map. But if \(X'\) is the spectrum of a regular local ring then \(\mathcal{L}\) is trivial by the description of the Picard group in Lemma 0C5G. Thus the claim is proved.

Combining the claims proved in the previous two paragraphs we see that the map \(\mathcal{L} \mapsto b_*\mathcal{L}\) is an inverse to the map \[\Pic(X') \longrightarrow \Ker(\Pic(X) \to \Pic(E))\] and the lemma is proved.

Remark

Let \(b : X \to X'\) be the contraction of an exceptional curve of the first kind \(E \subset X\). From Lemma 0C5L we obtain an identification \[\Pic(X) = \Pic(X') \oplus \mathbf{Z}\] where \(\mathcal{L}\) corresponds to the pair \((\mathcal{L}', n)\) if and only if \(\mathcal{L} = (b^*\mathcal{L}')(-nE)\), i.e., \(\mathcal{L}(nE) = b^*\mathcal{L}'\). In fact the proof of Lemma 0C5L shows that \(\mathcal{L}' = b_*\mathcal{L}(nE)\). Of course the assignment \(\mathcal{L} \mapsto \mathcal{L}'\) is a group homomorphism.

Lemma

Let \(X\) be a Noetherian scheme. Let \(E \subset X\) be an exceptional curve of the first kind. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \(n\) be the integer such that \(\mathcal{L}|_E\) has degree \(n\) viewed as an invertible module on \(\mathbf{P}^1\). Then

1. If \(H^1(X, \mathcal{L}) = 0\) and \(n \geq 0\), then \(H^1(X, \mathcal{L}(iE)) = 0\) for \(0 \leq i \leq n + 1\).

2. If \(n \leq 0\), then \(H^1(X, \mathcal{L}) \subset H^1(X, \mathcal{L}(E))\).

Proof

Observe that \(\mathcal{L}|_E = \mathcal{O}(n)\) by Divisors, Lemma 0BXJ. Use induction, the long exact cohomology sequence associated to the short exact sequence \[0 \to \mathcal{L} \to \mathcal{L}(E) \to \mathcal{L}(E)|_E \to 0,\] and use the fact that \(H^1(\mathbf{P}^1, \mathcal{O}(d)) = 0\) for \(d \geq -1\) and \(H^0(\mathbf{P}^1, \mathcal{O}(d)) = 0\) for \(d \leq -1\). Some details omitted.

Lemma

Let \(S = \Spec(R)\) be an affine Noetherian scheme. Let \(X \to S\) be a proper morphism. Let \(\mathcal{L}\) be an ample invertible sheaf on \(X\). Let \(E \subset X\) be an exceptional curve of the first kind. Then

1. there exists a contraction \(b : X \to X'\) of \(E\),

2. \(X'\) is proper over \(S\), and

3. the invertible \(\mathcal{O}_{X'}\)-module \(\mathcal{L}'\) is ample with \(\mathcal{L}'\) as in Remark 0C5M.

Proof

Let \(n\) be the degree of \(\mathcal{L}|_E\) as in Lemma 0C2J. Observe that \(n > 0\) as \(\mathcal{L}\) is ample on \(E\) (Varieties, Lemma 0B5X and Properties, Lemma 01PU). After replacing \(\mathcal{L}\) by a power we may assume \(H^i(X, \mathcal{L}^{\otimes e}) = 0\) for all \(i > 0\) and \(e > 0\), see Cohomology of Schemes, Lemma 0B5U. Finally, after replacing \(\mathcal{L}\) by another power we may assume there exist global sections \(t_0, \ldots, t_n\) of \(\mathcal{L}\) which define a closed immersion \(\psi : X \to \mathbf{P}^n_S\), see Morphisms, Lemma 01VT.

Set \(\mathcal{M} = \mathcal{L}(nE)\). Then \(\mathcal{M}|_E \cong \mathcal{O}_E\). Since we have the short exact sequence \[0 \to \mathcal{M}(-E) \to \mathcal{M} \to \mathcal{O}_E \to 0\] and since \(H^1(X, \mathcal{M}(-E))\) is zero (by Lemma 0C2J and the fact that \(n > 0\)) we can pick a section \(s_{n + 1}\) of \(\mathcal{M}\) which generates \(\mathcal{M}|_E\). Finally, denote \(s_0, \ldots, s_n\) the sections of \(\mathcal{M}\) we get from the sections \(t_0, \ldots, t_n\) of \(\mathcal{L}\) chosen above via \(\mathcal{L} \subset \mathcal{L}(nE) = \mathcal{M}\). Combined the sections \(s_0, \ldots, s_n, s_{n + 1}\) generate \(\mathcal{M}\) in every point of \(X\) and therefore define a morphism \[\varphi : X \longrightarrow \mathbf{P}^{n + 1}_S\] over \(S\), see Constructions, Lemma 01NE.

Below we will check the conditions of Lemma 0C2L. Once this is done we see that the Stein factorization \(X \to X' \to \mathbf{P}^{n + 1}_S\) of \(\varphi\) is the desired contraction which proves (1). Moreover, the morphism \(X' \to \mathbf{P}^{n + 1}_S\) is finite hence \(X'\) is proper over \(S\) (Morphisms, Lemmas 01WN and 01W3). This proves (2). Observe that \(X'\) has an ample invertible sheaf. Namely the pullback \(\mathcal{M}'\) of \(\mathcal{O}_{\mathbf{P}^{n + 1}_S}(1)\) is ample by Morphisms, Lemma 0892. Observe that \(\mathcal{M}'\) pulls back to \(\mathcal{M}\) on \(X\) (by Constructions, Lemma 01NE). Finally, \(\mathcal{M} = \mathcal{L}(nE)\). Since in the arguments above we have replaced the original \(\mathcal{L}\) by a positive power we conclude that the invertible \(\mathcal{O}_{X'}\)-module \(\mathcal{L}'\) mentioned in (3) of the lemma is ample on \(X'\) by Properties, Lemma 01PT.

Easy observations: \(\mathbf{P}^{n + 1}_S\) is Noetherian and \(\varphi\) is proper. Details omitted.

Next, we observe that any point of \(U = X \setminus E\) is mapped to the open subscheme \(W\) of \(\mathbf{P}^{n + 1}_S\) where one of the first \(n + 1\) homogeneous coordinates is nonzero. On the other hand, any point of \(E\) is mapped to a point where the first \(n + 1\) homogeneous coordinates are all zero, in particular into the complement of \(W\). Moreover, it is clear that there is a factorization \[U = \varphi^{-1}(W) \xrightarrow{\varphi|_U} W \xrightarrow{\text{pr}} \mathbf{P}^n_S\] of \(\psi|_U\) where \(\text{pr}\) is the projection using the first \(n + 1\) coordinates and \(\psi : X \to \mathbf{P}^n_S\) is the embedding chosen above. It follows that \(\varphi|_U : U \to W\) is quasi-finite.

Finally, we consider the map \(\varphi|_E : E \to \mathbf{P}^{n + 1}_S\). Observe that for any point \(x \in E\) the image \(\varphi(x)\) has its first \(n + 1\) coordinates equal to zero, i.e., the morphism \(\varphi|_E\) factors through the closed subscheme \(\mathbf{P}^0_S \cong S\). The morphism \(E \to S = \Spec(R)\) factors as \(E \to \Spec(H^0(E, \mathcal{O}_E)) \to \Spec(R)\) by Schemes, Lemma 01I1. Since by assumption \(H^0(E, \mathcal{O}_E)\) is a field we conclude that \(E\) maps to a point in \(S \subset \mathbf{P}^{n + 1}_S\) which finishes the proof.

Lemma

Let \(S\) be a Noetherian scheme. Let \(f : X \to S\) be a morphism of finite type. Let \(E \subset X\) be an exceptional curve of the first kind which is in a fibre of \(f\).

1. If \(X\) is projective over \(S\), then there exists a contraction \(X \to X'\) of \(E\) and \(X'\) is projective over \(S\).

2. If \(X\) is quasi-projective over \(S\), then there exists a contraction \(X \to X'\) of \(E\) and \(X'\) is quasi-projective over \(S\).

Proof

Both cases follow from Lemma 0C2M using standard results on ample invertible modules and (quasi-)projective morphisms.

Proof of (1). Projectivity of \(f\) means that \(f\) is proper and there exists an \(f\)-ample invertible module \(\mathcal{L}\), see Morphisms, Lemma 0BCL and Definition 01VW. Let \(U \subset S\) be an affine open containing the image of \(E\). By Lemma 0C2M there exists a contraction \(c : f^{-1}(U) \to V'\) of \(E\) and an ample invertible module \(\mathcal{N}'\) on \(V'\) whose pullback to \(f^{-1}(U)\) is equal to \(\mathcal{L}(nE)|_{f^{-1}(U)}\). Let \(v \in V'\) be the closed point such that \(c\) is the blowing up of \(v\). Then we can glue \(V'\) and \(X \setminus E\) along \(f^{-1}(U) \setminus E = V' \setminus \{v\}\) to get a scheme \(X'\) over \(S\). The morphisms \(c\) and \(\text{id}_{X \setminus E}\) glue to a morphism \(b : X \to X'\) which is the contraction of \(E\). The inverse image of \(U\) in \(X'\) is proper over \(U\). On the other hand, the restriction of \(X' \to S\) to the complement of the image of \(v\) in \(S\) is isomorphic to the restriction of \(X \to S\) to that open. Hence \(X' \to S\) is proper (as being proper is local on the base by Morphisms, Lemma 01W2). Finally, \(\mathcal{N}'\) and \(\mathcal{L}|_{X \setminus E}\) restrict to isomorphic invertible modules over \(f^{-1}(U) \setminus E = V' \setminus \{v\}\) and hence glue to an invertible module \(\mathcal{L}'\) over \(X'\). The restriction of \(\mathcal{L}'\) to the inverse image of \(U\) in \(X'\) is ample because this is true for \(\mathcal{N}'\). For affine opens of \(S\) avoiding the image of \(v\), we see that the same is true because it holds for \(\mathcal{L}\). Thus \(\mathcal{L}'\) is \((X' \to S)\)-relatively ample by Morphisms, Lemma 01VJ and (1) is proved.

Proof of (2). We can write \(X\) as an open subscheme of a scheme \(\overline{X}\) projective over \(S\) by Morphisms, Lemma 07RM. By (1) there is a contraction \(b : \overline{X} \to \overline{X}'\) and \(\overline{X}'\) is projective over \(S\). Then we let \(X' \subset \overline{X}\) be the image of \(X \to \overline{X}'\); this is an open as \(b\) is an isomorphism away from \(E\). Then \(X \to X'\) is the desired contraction. Note that \(X'\) is quasi-projective over \(S\) as it has an \(S\)-relatively ample invertible module by the construction in the proof of part (1).

Lemma

Let \(S\) be a Noetherian scheme. Let \(f : X \to S\) be a separated morphism of finite type with \(X\) regular of dimension \(2\). Then \(X\) is quasi-projective over \(S\).

Proof

By Chow’s lemma (Cohomology of Schemes, Lemma 0200) there exists a proper morphism \(\pi : X' \to X\) which is an isomorphism over a dense open \(U \subset X\) such that \(X' \to S\) is H-quasi-projective. By Lemma 0C5H there exists a sequence of blowups in closed points \[X_n \to \ldots \to X_1 \to X_0 = X\] and an \(S\)-morphism \(X_n \to X'\) extending the rational map \(U \to X'\). Observe that \(X_n \to X\) is projective by Divisors, Lemma 02NS and Morphisms, Lemma 0C4P. This implies that \(X_n \to X'\) is projective by Morphisms, Lemma 0C4Q. Hence \(X_n \to S\) is quasi-projective by Morphisms, Lemma 0C4M (and the fact that a projective morphism is quasi-projective, see Morphisms, Lemma 07RL). By Lemma 0C2N (and uniqueness of contractions Lemma 0C5K) we conclude that \(X_{n - 1}, \ldots, X_0 = X\) are quasi-projective over \(S\) as desired.

Lemma

Let \(S\) be a Noetherian scheme. Let \(f : X \to S\) be a proper morphism with \(X\) regular of dimension \(2\). Then \(X\) is projective over \(S\).

Proof

This follows from Lemma 0C5N and Morphisms, Lemma 0BCL.

Factorization birational maps

Proper birational morphisms between nonsingular surfaces are given by sequences of quadratic transforms.

Lemma

Let \(f : X \to Y\) be a proper birational morphism between integral Noetherian schemes regular of dimension \(2\). Then \(f\) is a sequence of blowups in closed points.

Proof

Let \(V \subset Y\) be the maximal open over which \(f\) is an isomorphism. Then \(V\) contains all codimension \(1\) points of \(V\) (Varieties, Lemma 0BFP). Let \(y \in Y\) be a closed point not contained in \(V\). Then we want to show that \(f\) factors through the blowup \(b : Y' \to Y\) of \(Y\) at \(y\). Namely, if this is true, then at least one (and in fact exactly one) component of the fibre \(f^{-1}(y)\) will map isomorphically onto the exceptional curve in \(Y'\) and the number of curves in fibres of \(X \to Y'\) will be strictly less that the number of curves in fibres of \(X \to Y\), so we conclude by induction. Some details omitted.

By Lemma 0C5H we know that there exists a sequence of blowing ups \[X' = X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X\] in closed points lying over the fibre \(f^{-1}(y)\) and a morphism \(X' \to Y'\) such that \[\xymatrix{ X' \ar[d]_{f'} \ar[r] & X \ar[d]^f \\ Y' \ar[r] & Y }\] is commutative. We want to show that the morphism \(X' \to Y'\) factors through \(X\) and hence we can use induction on \(n\) to reduce to the case where \(X' \to X\) is the blowup of \(X\) in a closed point \(x \in X\) mapping to \(y\).

Let \(E \subset X'\) be the exceptional fibre of the blowing up \(X' \to X\). If \(E\) maps to a point in \(Y'\), then we obtain the desired factorization by Lemma 0C5J. We will prove that if this is not the case we obtain a contradiction. Namely, if \(f'(E)\) is not a point, then \(E' = f'(E)\) must be the exceptional curve in \(Y'\). Picture \[\xymatrix{ E \ar[r] \ar[d]_g & X' \ar[d]_{f'} \ar[r] & X \ar[d]^f \\ E' \ar[r] & Y' \ar[r] & Y }\] Arguing as before \(f'\) is an isomorphism in an open neighbourhood of the generic point of \(E'\). Hence \(g : E \to E'\) is a finite birational morphism. Then the inverse of \(g\) (a rational map) is everywhere defined by Morphisms, Lemma 0BX7 and \(g\) is an isomorphism. Consider the map \[g^*\mathcal{C}_{E'/Y'} \longrightarrow \mathcal{C}_{E/X'}\] of Morphisms, Lemma 01R4. Since the source and target are invertible modules of degree \(1\) on \(E = E' = \mathbf{P}^1_\kappa\) and since the map is nonzero (as \(f'\) is an isomorphism in the generic point of \(E\)) we conclude it is an isomorphism. By Morphisms, Lemma 067L we conclude that \(\Omega_{X'/Y'}|_E = 0\). This means that \(f'\) is unramified at every point of \(E\) (Morphisms, Lemma 02GF). Hence \(f'\) is quasi-finite at every point of \(E\) (Morphisms, Lemma 02V5). Hence the maximal open \(V' \subset Y'\) over which \(f'\) is an isomorphism contains \(E'\) by Varieties, Lemma 0BFP. This in turn implies that the inverse image of \(y\) in \(X'\) is \(E'\). Hence the inverse image of \(y\) in \(X\) is \(x\). Hence \(x \in X\) is in the maximal open over which \(f\) is an isomorphism by Varieties, Lemma 0BFP. This is a contradiction as we assumed that \(y\) is not in this open.

Lemma

Let \(S\) be a Noetherian scheme. Let \(X\) and \(Y\) be proper integral schemes over \(S\) which are regular of dimension \(2\). Then \(X\) and \(Y\) are \(S\)-birational if and only if there exists a diagram of \(S\)-morphisms \[X = X_0 \leftarrow X_1 \leftarrow \ldots \leftarrow X_n = Y_m \to \ldots \to Y_1 \to Y_0 = Y\] where each morphism is a blowup in a closed point.

Proof

Let \(U \subset X\) be open and let \(f : U \to Y\) be the given \(S\)-rational map (which is invertible as an \(S\)-rational map). By Lemma 0C5H we can factor \(f\) as \(X_n \to \ldots \to X_1 \to X_0 = X\) and \(f_n : X_n \to Y\). Since \(X_n\) is proper over \(S\) and \(Y\) separated over \(S\) the morphism \(f_n\) is proper. Clearly \(f_n\) is birational. Hence \(f_n\) is a composition of contractions by Lemma 0C5R. We omit the proof of the converse.