Logarithmic derivatives and going up

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

The previous lesson showed that every blow-up sequence of order \(m\) for an ideal restricts to a hypersurface of maximal contact, but that the restriction may demand extra blow-ups. This lesson proves the converse for a class of ideals, the D-balanced ones, whose derivatives are controlled by the ideal itself: for such an ideal, every blow-up sequence of the restriction to any smooth hypersurface, pushed forward, is a blow-up sequence for the ideal. The proof separates derivatives into those tangent to the hypersurface, which commute with restriction and with blowing up, and the remaining normal direction. The tangent ones are the logarithmic derivations.

We use Smooth blow-ups and transforms of ideals, Blow-up sequences and the main theorems, Derivative ideals under blowing up and Hypersurfaces of maximal contact.

1. D-balanced marked ideals

Definition 1.1. A marked ideal \((\mathcal I,m)\), \(m\ge1\), is D-balanced if

\[ \bigl(D^i\mathcal I\bigr)^m\subset\mathcal I^{m-i}\qquad\text{for }0\le i<m. \]

An ideal \(\mathcal I\) with \(m=\operatorname{maxord}\mathcal I\) is D-balanced if \((\mathcal I,m)\) is.

Lemma 1.2. If \((\mathcal I,m)\) is D-balanced, then at every point \(x\) either \(\mathcal I_x=\mathcal O_{X,x}\) or \(\operatorname{ord}_x\mathcal I\ge m\). Hence \(\operatorname{cosupp}(\mathcal I,m)=V(\mathcal I)\), and for every smooth hypersurface \(S\), \(\operatorname{cosupp}(\mathcal I|_S,m)=S\cap\operatorname{cosupp}(\mathcal I,m)\).

Proof. If \(\operatorname{ord}_x\mathcal I=a<m\), then \(D^a(\mathcal I)\), hence \(D^{m-1}(\mathcal I)\), contains a unit at \(x\) by the Taylor criterion. Then \((D^{m-1}\mathcal I)^m\subset\mathcal I\) contains a unit. For the restriction, orders do not decrease under restriction to \(S\), and where \(\mathcal I_x=\mathcal O_{X,x}\) also \((\mathcal I|_S)_x=\mathcal O_{S,x}\). \(\square\)

So for a single blow-up with centre in \(S\), being of order \(\ge m\) for \((\mathcal I|_S,m)\) and for \((\mathcal I,m)\) are the same condition. The work is to keep this true along whole sequences, although the transforms of a D-balanced ideal need not be D-balanced (Example 5.1).

2. Logarithmic derivations along a hypersurface

Let \(S\subset X\) be a smooth hypersurface.

Definition 2.1. \(\operatorname{Der}_X(-\log S)\) is the sheaf of derivations \(\delta\) with \(\delta(\mathcal I_S)\subset\mathcal I_S\). For an ideal \(\mathcal I\), \(D(-\log S)(\mathcal I)\) is the ideal generated by \(\mathcal I\) and by \(\delta(f)\), \(\delta\in\operatorname{Der}_X(-\log S)\), \(f\in\mathcal I\); and \(D^{r+1}(-\log S)=D(-\log S)\circ D^r(-\log S)\).

Lemma 2.2. Near a point of \(S\), take coordinates with \(S=V(z_1)\) (possible by Lemma 1.2 of the first lesson). Then \(\operatorname{Der}_X(-\log S)\) is generated by \(z_1\partial_1,\partial_2,\ldots,\partial_N\). These generators commute, and \(D^r(-\log S)(\mathcal I)\) is generated by the \((z_1\partial_1)^a\partial^{\alpha'}g_l\) with \(a+|\alpha'|\le r\), where \(g_l\) are generators of \(\mathcal I\) and \(\alpha'\) is supported in \(\{2,\ldots,N\}\). Likewise \(D^r(\mathcal I)\) is generated by the \(\partial^\alpha g_l\), \(|\alpha|\le r\).

Proof. A derivation \(\delta=\sum a_i\partial_i\) preserves \((z_1)\) if and only if \(\delta(z_1)=a_1\in(z_1)\), because \(\delta(z_1g)=g\,\delta(z_1)+z_1\delta(g)\). Writing \(a_1=z_1b\) gives \(\delta=b\,z_1\partial_1+\sum_{i\ge2}a_i\partial_i\). The operators \(z_1\partial_1\) and \(\partial_i\), \(i\ge2\), commute because \(\partial_iz_1=0\). The description of the iterated ideals follows as for \(D^r\) in Smooth blow-ups and transforms of ideals, Definition 2.4, using the Leibniz rule to move derivatives off the coefficients: the derivative of \(a\,g_l\) is a combination of derivatives of \(g_l\) of the same or lower order. \(\square\)

Proposition 2.3. Let \(S=V(z_1)\) as in Lemma 2.2, and write \((\partial_1^j\mathcal I)\) for the ideal generated by the \(\partial_1^jg_l\).

  1. Restriction. \(\bigl(D^r(-\log S)(\mathcal I)\bigr)|_S=D^r(\mathcal I|_S)\).
  2. Decomposition. \(D^s(\mathcal I)=\sum_{j=0}^sD^{s-j}(-\log S)\bigl((\partial_1^j\mathcal I)\bigr)\).

Proof. (1) The derivations \(\partial_i\), \(i\ge2\), preserve \((z_1)\) and induce the coordinate derivations of \(S\) for the coordinates \(z_2|_S,\ldots,z_N|_S\). So \((\partial^{\alpha'}g)|_S=\partial^{\alpha'}(g|_S)\). The operator \(z_1\partial_1\) maps every function into \((z_1)\), and so does every operator of the form \((z_1\partial_1)^a\partial^{\alpha'}\) with \(a\ge1\), since \(z_1\partial_1\) preserves \((z_1)\). Hence the generators of Lemma 2.2 restrict to \(0\) or to the generators \(\partial^{\alpha'}(g_l|_S)\) of \(D^r(\mathcal I|_S)\).

(2) The right side is contained in the left, since logarithmic derivations are derivations and \(\partial_1^jg_l\in D^j\mathcal I\). Conversely \(D^s(\mathcal I)\) is generated by the \(\partial^\alpha g_l=\partial^{\alpha'}(\partial_1^{a}g_l)\) with \(a+|\alpha'|\le s\), and each lies in \(D^{|\alpha'|}(-\log S)((\partial_1^a\mathcal I))\). \(\square\)

Proposition 2.4 (logarithmic derivatives under blowing up). Let \((X_r,\mathcal I_r,m)\to\cdots\to(X_0,\mathcal I_0,m)\) be a smooth blow-up sequence of order \(\ge m\), and \(S\subset X\) a smooth hypersurface whose strict transforms \(S_i\) contain the centres \(Z_i\), no \(Z_i\) containing a component of \(S_i\). Then for \(0\le j\le m\) the sequence is of order \(\ge m-j\) for \(\bigl(D^j(-\log S)(\mathcal I),m-j\bigr)\), and

\[ \Pi^{-1}_*\bigl(D^j(-\log S)(\mathcal I),\,m-j\bigr)\subset D^j(-\log S_r)\bigl(\Pi^{-1}_*(\mathcal I,m)\bigr). \]

Proof. The orders are fine because \(D^j(-\log S)(\mathcal I)\subset D^j(\mathcal I)\) and Derivative ideals under blowing up, Theorem 3.1 applies. As there, it suffices to treat one blow-up and \(j=1\); higher \(j\) and longer sequences follow by the same two inductions. Take coordinates with \(S=V(z_1)\) and \(Z=V(z_1,\ldots,z_r)\), \(r\ge2\) since \(Z\) contains no component of \(S\) (Lemma 1.3 of the first lesson). On the chart \(U_1\) the strict transform \(S_1\) is empty, \(\operatorname{Der}(-\log\emptyset)\) consists of all derivations, and the claim is the case \(j=1\) of that Theorem 3.1. On \(U_j\), \(2\le j\le r\), \(S_1=V(y_1)\), and \(\operatorname{Der}(-\log S_1)\) is generated by \(y_1\partial/\partial y_1\) and \(\partial/\partial y_i\), \(i\ge2\). The generators of \(D(-\log S)(\mathcal I)\) are \(f\), \(z_1\partial_1f\) and \(\partial_if\), \(i\ge2\), for \(f\in\mathcal I\). By Derivative ideals under blowing up, Proposition 2.1, with \(G\) the transform of \((f,m)\):

Every right side lies in \(D(-\log S_1)(G)\). \(\square\)

3. Comparing derivatives after blowing up

Theorem 3.1. In the situation of Proposition 2.4, for every \(s\le m\),

\[ D^s\bigl(\Pi^{-1}_*(\mathcal I,m)\bigr)=\sum_{j=0}^sD^{s-j}(-\log S_r)\Bigl(\Pi^{-1}_*\bigl(D^j\mathcal I,\,m-j\bigr)\Bigr). \]

Proof. The right side is contained in the left: logarithmic derivatives are derivatives, and by Derivative ideals under blowing up, Theorem 3.1, \(D^{s-j}\Pi^{-1}_*(D^j\mathcal I,m-j)\subset D^{s-j}D^j\Pi^{-1}_*(\mathcal I,m)=D^s\Pi^{-1}_*(\mathcal I,m)\).

One blow-up. On \(U_1\), \(S_1=\emptyset\) and the term \(j=0\) is already \(D^s\pi^{-1}_*(\mathcal I,m)\). On \(U_j\), \(j\ge2\), apply Proposition 2.3(2) to the ideal \(\pi^{-1}_*(\mathcal I,m)\), generated by the transforms \(G_l\) of generators \(g_l\) of \(\mathcal I\), and the hypersurface \(S_1=V(y_1)\):

\[ D^s\pi^{-1}_*(\mathcal I,m)=\sum_{j=0}^sD^{s-j}(-\log S_1)\bigl((\partial_{y_1}^jG_l)_l\bigr). \]

By the first formula of Proposition 2.1 of the previous lesson, applied \(j\) times (each time to a marked function of marking one less), \(\partial_{y_1}^jG_l\) represents \(\pi^{-1}_*(\partial_1^jg_l,m-j)\), which lies in \(\pi^{-1}_*(D^j\mathcal I,m-j)\). This proves the reverse inclusion for one blow-up.

Induction on the length. Factor \(\Pi=\Pi_{r-1}\circ\pi_{r-1}\). The one-blow-up case for \(\pi_{r-1}\) and the marked ideal \(\Pi_{r-1}{}^{-1}_*(\mathcal I,m)\) gives

\[ D^s\Pi^{-1}_*(\mathcal I,m)=\sum_{j}D^{s-j}(-\log S_r)\,(\pi_{r-1})^{-1}_*\bigl(D^j\Pi_{r-1}{}^{-1}_*(\mathcal I,m),\,m-j\bigr). \]

By induction, \(D^j\Pi_{r-1}{}^{-1}_*(\mathcal I,m)=\sum_{\ell\le j}D^{j-\ell}(-\log S_{r-1})\Pi_{r-1}{}^{-1}_*(D^\ell\mathcal I,m-\ell)\). Proposition 2.4 for the single blow-up \(\pi_{r-1}\) moves \((\pi_{r-1})^{-1}_*\) past \(D^{j-\ell}(-\log S_{r-1})\), turning it into \(D^{j-\ell}(-\log S_r)\). Substituting and using \(D^{a}(-\log S_r)D^{b}(-\log S_r)=D^{a+b}(-\log S_r)\) gives the right side of the theorem as an upper bound. \(\square\)

Corollary 3.2. In the situation of Proposition 2.4, assume also that \(\mathcal I|_S\) is nonzero on every component of \(S\). Write \(\Pi^S\) for the restricted sequence on \(S\). Then

\[ S_r\cap\operatorname{cosupp}\bigl(\Pi^{-1}_*(\mathcal I,m)\bigr)=\bigcap_{j=0}^{m-1}\operatorname{cosupp}\Bigl((\Pi^S)^{-1}_*\bigl((D^j\mathcal I)|_S,\,m-j\bigr)\Bigr), \]

where the transforms on the right are the restrictions to \(S_r\) of the transforms of \((D^j\mathcal I,m-j)\) along \(\Pi\).

Proof. By Smooth blow-ups and transforms of ideals, Lemma 4.4, applied step by step, restriction to \(S_i\) commutes with the marked transforms of \(\mathcal I\) and of each \(D^j\mathcal I\supset\mathcal I\); these restrictions stay nonzero on every component, as in the proof of Hypersurfaces of maximal contact, Theorem 2.3(3). Restrict the identity of Theorem 3.1 with \(s=m-1\) to \(S_r\). By Proposition 2.3(1) and Lemma 4.4, the right side becomes \(\sum_jD^{m-1-j}\bigl((\Pi^S)^{-1}_*((D^j\mathcal I)|_S,m-j)\bigr)\). Now take the vanishing loci. For every ideal \(\mathcal K\), \(V(\mathcal K|_S)=V(\mathcal K)\cap S\). On the left, \(V(D^{m-1}\Pi^{-1}_*(\mathcal I,m))=\operatorname{cosupp}(\Pi^{-1}_*(\mathcal I,m),m)\) by Corollary 2.5 of the first lesson. On the right, the vanishing locus of a sum is the intersection of the vanishing loci, and \(V(D^{m-1-j}\mathcal T)=\operatorname{cosupp}(\mathcal T,m-j)\) on \(S_r\) for each term. \(\square\)

4. Going up

Theorem 4.1 (going up). Let \((\mathcal I,m)\) be D-balanced, and let \(S\subset X\) be a smooth hypersurface none of whose components lies in \(\operatorname{cosupp}(\mathcal I,m)\). Then the push-forward of every smooth blow-up sequence of order \(\ge m\) starting with \((S,\mathcal I|_S,m)\) is a smooth blow-up sequence of order \(\ge m\) starting with \((X,\mathcal I,m)\), whose transforms restrict to the transforms on \(S\). If \(m=\operatorname{maxord}\mathcal I\), it is a blow-up sequence of order \(m\) for \((X,\mathcal I)\).

Proof. By Lemma 1.2, \(\mathcal I|_S\) is nonzero on every component of \(S\). Let \((S_i,\mathcal J_i,m)\) be the sequence on \(S\) with centres \(Z_i\subset S_i\); no \(Z_i\) contains a component of \(S_i\), since \(\mathcal J_i\) is nonzero on every component. We show by induction on \(i\) that the push-forward up to step \(i\) is a sequence of order \(\ge m\) for \((X,\mathcal I,m)\) with \(\mathcal I_i|_{S_i}=\mathcal J_i\). For \(i=0\) this is clear. Suppose it holds for \(i=r-1\). The restriction identity at step \(r\) follows from Lemma 4.4 of the first lesson once the order condition holds, so we must show \(Z_{r-1}\subset\operatorname{cosupp}(\mathcal I_{r-1},m)\). By Corollary 3.2 for the first \(r-1\) steps,

\[ \begin{aligned} S_{r-1}\cap\operatorname{cosupp}(\mathcal I_{r-1},m)&=\bigcap_{j<m}\operatorname{cosupp}\,(\Pi^S)^{-1}_*\bigl((D^j\mathcal I)|_S,\,m-j\bigr)\\ &=\bigcap_{j<m}\operatorname{cosupp}\,(\Pi^S)^{-1}_*\bigl((D^j\mathcal I)^m|_S,\,m(m-j)\bigr)\\ &\supset\bigcap_{j<m}\operatorname{cosupp}\,(\Pi^S)^{-1}_*\bigl(\mathcal I^{m-j}|_S,\,m(m-j)\bigr)\\ &=\operatorname{cosupp}\,(\Pi^S)^{-1}_*\bigl(\mathcal I|_S,\,m\bigr)=\operatorname{cosupp}(\mathcal J_{r-1},m). \end{aligned} \]

The second line uses that transforms commute with powers and \(\operatorname{cosupp}(\mathcal K^c,cw)=\operatorname{cosupp}(\mathcal K,w)\). The third uses \((D^j\mathcal I)^m\subset\mathcal I^{m-j}\): a smaller ideal has a smaller transform and a larger cosupport. All transforms involved are restrictions of transforms along the push-forward, which is a sequence of order \(\ge m\) for \((\mathcal I,m)\) up to step \(r-1\); so they are defined by Theorem 3.1 of the previous lesson and Lemma 4.4. Since \(Z_{r-1}\subset\operatorname{cosupp}(\mathcal J_{r-1},m)\), the induction step is complete. The last statement follows from Lemma 2.3 of Blow-up sequences and the main theorems. \(\square\)

Lemma 4.2 (snc and a transversal hypersurface). Let \(H\) be a smooth hypersurface and \(E\) an snc divisor such that \((H,E^1,\ldots,E^s)\) is snc. A smooth centre \(Z\subset H\) has snc with \(E\) in \(X\) if and only if it has snc with \(E|_H\) in \(H\). In that case the total transform of \(E\) restricts on the strict transform \(B_ZH\) to the total transform of \(E|_H\), and \((B_ZH,\pi^{-1}_{\rm tot}E)\) is snc.

Proof. If coordinates on \(X\) are adapted to \(H=V(z_1)\), \(E\) and \(Z\), their restrictions to \(H\) are adapted to \(E|_H\) and \(Z\). Conversely, let \(w\) be coordinates on \(H\) near a closed point \(p\), centred at \(p\), adapted to \(E|_H\) and \(Z\), with \(E^i\cap H=V(w_{c(i)})\) for the \(E^i\) through \(p\). If \(e_i\) is a local equation of \(E^i\), then \(e_i|_H=u_iw_{c(i)}\) with \(u_i\) a unit, and replacing \(w_{c(i)}\) by \(e_i|_H\) keeps a coordinate system centred at \(p\), adapted to \(Z\) because \(V(u_iw_{c(i)})=V(w_{c(i)})\). Lift the \(w\) to functions on \(X\), using \(e_i\) as the lift of \(e_i|_H\), and add a local equation of \(H\). As in the proof of Lemma 1.3 of the first lesson, this is a coordinate system on \(X\), adapted to \(H\), \(E\) and \(Z\). The statement on total transforms follows in these coordinates from the chart description of Smooth blow-ups and transforms of ideals, Lemma 3.3. \(\square\)

Corollary 4.3 (going up and down). Let \(\mathcal I\) be a D-balanced ideal with \(m=\operatorname{maxord}\mathcal I\), \(E\) an snc divisor, and \(H\) an MC-hypersurface for \(\mathcal I\) with \((H,E)\) snc and no component of \(H\) in \(\operatorname{cosupp}(\mathcal I,m)\). Then push-forward from \(H\) is a bijection between blow-up sequences of order \(\ge m\) starting with \((H,\mathcal I|_H,m,E|_H)\) and blow-up sequences of order \(m\) starting with \((X,\mathcal I,E)\), compatible with restricting the transforms to \(H\).

Proof. By Lemma 1.2, \(\mathcal I|_H\) is nonzero on every component of \(H\). Going down (Hypersurfaces of maximal contact, Theorem 2.3(3)) and going up (Theorem 4.1) give inverse maps between the sequences without divisors. By Lemma 4.2 the snc conditions on the two sides agree at every step, and the total transforms correspond. The strict transforms \(H_i\) stay transversal to the total transforms of \(E\), because every centre lies in \(H_i\) and has snc with \(H_i+E_i\). \(\square\)

5. Examples

Example 5.1 (transforms need not stay D-balanced). Let \(\mathcal I=(x^2,xy^n,y^{n+1})\) on \(\mathbf A^2\) with \(n\ge2\), marked with \(2\). Then \(D(\mathcal I)=(x,y^n)\) and \(D(\mathcal I)^2=(x^2,xy^n,y^{2n})\subset\mathcal I\), so \((\mathcal I,2)\) is D-balanced. Blow up the origin; on the chart \(x=x_1y\) the transform is \(\mathcal I_1=(x_1^2,y^{n-1})\). Now \(D(\mathcal I_1)=(x_1,y^{n-2})\), and \(x_1y^{n-2}\in D(\mathcal I_1)^2\) is not in \(\mathcal I_1\). So \((\mathcal I_1,2)\) is not D-balanced. Theorem 4.1 avoids this problem by working with \(\mathcal I\) and its derivatives on \(X\), never with the D-balance of the transforms.

Example 5.2 (adding derivatives removes the obstruction of Example 3.1 in Hypersurfaces of maximal contact). For \(\mathcal I=(xy-z^n)\), \(n\ge2\), consider \(\mathcal K=\mathcal I+D(\mathcal I)^2\). Since \(D(\mathcal I)=(x,y,z^{n-1})\) and \(xy\in D(\mathcal I)^2\), also \(z^n\in\mathcal K\), and

\[ \mathcal K=(x^2,xy,y^2,xz^{n-1},yz^{n-1},z^n). \]

Its maximal order is \(2\), \(D(\mathcal K)=(x,y,z^{n-1})\), and \(D(\mathcal K)^2\subset\mathcal K\), so \(\mathcal K\) is D-balanced. Its restriction to \(S=V(x)\) is \((y^2,yz^{n-1},z^n)\), which has the single point of order \(2\) at the origin. By Theorem 4.1, every blow-up sequence of order \(\ge2\) for this restriction pushes forward to one of order \(2\) for \(\mathcal K\). The relation between order reduction for \(\mathcal K\) and for \(\mathcal I\) is the subject of the lesson on tuning.

6. Exercises

Exercise 6.1. Show that \(\operatorname{Der}_{\mathbf A^2}(-\log V(xy))\), the derivations preserving both \((x)\) and \((y)\), is generated by \(x\partial_x\) and \(y\partial_y\).

Solution. \(\delta=a\partial_x+b\partial_y\) preserves \((x)\) if and only if \(a\in(x)\), and \((y)\) if and only if \(b\in(y)\). So \(\delta=a'x\partial_x+b'y\partial_y\).

Exercise 6.2. Verify Proposition 2.3(1) for \(\mathcal I=(x^2+xy+z^3)\), \(S=V(x)\) and \(r=1\), and compare with \(D(\mathcal I)|_S\).

Solution. \(D(-\log S)(\mathcal I)\) is generated by \(f=x^2+xy+z^3\), \(x\partial_xf=2x^2+xy\), \(\partial_yf=x\) and \(\partial_zf=3z^2\). Restricted to \(S\): \(z^3,0,0,3z^2\), giving \((z^2)=D(z^3)=D(\mathcal I|_S)\). By contrast \(D(\mathcal I)|_S\) also contains \(\partial_xf|_S=y\), so \(D(\mathcal I)|_S=(y,z^2)\ne D(\mathcal I|_S)\).

Exercise 6.3. In Example 5.1, check that the blow-up of the origin is of order \(\ge2\) for \((\mathcal I,2)\) and that \(\mathcal I_1\) has order \(2\) at the origin of the chart when \(n\ge3\).

Solution. The order of \(\mathcal I\) at the origin is \(2\), from \(x^2\). On the chart, \(\pi^*\mathcal I=y^2(x_1^2,x_1y^{n-1},y^{n-1})\), so \(\mathcal I_1=(x_1^2,y^{n-1})\), of order \(\min(2,n-1)=2\) for \(n\ge3\).

References