--- title: "Lecture 29 - Projections and Parametrised Projective Curves" stable_id: br-ak-2012-l29 language: en source_author: "Holger Brenner" frozen_revision_contributor: "Bocardodarapti" upstream_title: "Kurs:Algebraische Kurven (Osnabrück 2012)/Vorlesung 29" upstream_pageid: 51996 upstream_revid: 1069408 upstream_timestamp: "2026-02-05T19:18:37Z" upstream_mediawiki_sha1: 6f0742211aeb307841634425937aad9037da51be source_url: "https://de.wikiversity.org/w/index.php?oldid=1069408" authority_manifest: authority/wikiversity/unit-29/UNIT_AUTHORITY_MANIFEST.json authority_manifest_sha256: ec3b34ad387ae827ecaa365c4def3b0550f74b629d0db3873a7cc28dc0831bc5 lecture_xml: authority/wikiversity/unit-29/lecture-29.xml lecture_xml_sha256: e5055632a6aa8119540cb5acccc0ba86a82b6d2bc88192b9ddd5a77aaea31d70 lecture_expanded_tex: authority/wikiversity/unit-29/lecture-29-expanded.tex lecture_expanded_tex_sha256: 7c06a1dbb12904bd5f89427955ef8bdae5781e402522cd70f09a0c6e1ef1e784 license: "Current semantic course text and this translation: CC BY-SA 4.0. 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No blanket relicensing claim is made." source_component_license_route: "Semantic-site rights notice: CC BY-SA 4.0; media component rights remain item-specific; official-PDF notices remain component-specific; no blanket relicensing claim." license_evidence: "authority/UNIT_29_AUTHORITY_FREEZE.md; authority/RIGHTS-unit-29.csv; authority/ASSET_CLOSURE-unit-29.json" non_endorsement: "This independent English edition does not imply endorsement by the author, Wikiversity, the Wikimedia Foundation, or any source institution." translation_status: complete translation_provenance: "OpenAI Codex gpt-5.6-sol, Ultra." source_semantic_entities: 8 source_corrections: 7 correction_ids: "AGC-CORR-0126; AGC-CORR-0127; AGC-CORR-0128; AGC-CORR-0129; REVIEW-AK-26-30-C12; REVIEW-AK-26-30-C13; REVIEW-AK-26-30-C14" reader_media_positions: 0 --- # Lecture 29: Projections and Parametrised Projective Curves {#br-ak-2012-l29} ## Projection away from a point {#br-ak-2012-l29-s01} ### Definition 29.1: projection away from a point {#br-ak-2012-l29-def-01} The map $$ \begin{aligned} \mathbb P_K^n\setminus\{(1,0,\ldots,0)\} &\longrightarrow \mathbb P_K^{n-1},\\ (x_0,x_1,\ldots,x_n)&\longmapsto(x_1,\ldots,x_n) \end{aligned} $$ is called the *projection away from the point* $(1,0,\ldots,0)$. This map is a well-defined morphism outside the *centre* of projection $(1,0,\ldots,0)$. Each other point is sent to the point of $\mathbb P_K^{n-1}$ corresponding to the line through that point and the centre. Hence the map is surjective, and each fibre is a projective line with the centre removed, and thus an affine line. In other words, we have a so-called *line bundle* over $\mathbb P_K^{n-1}$. This map extends the cone map $$ \mathbb A_K^n\setminus\{0\}\longrightarrow\mathbb P_K^{n-1} $$ to punctured projective space. The corresponding map can be defined for any centre; see Exercise 29.7. ## Maps to $\mathbb P_K^1$ {#br-ak-2012-l29-s02} The following theorem gives a new version of Noether normalisation. ### Theorem 29.2: projecting a plane curve to $\mathbb P_K^1$ {#br-ak-2012-l29-thm-01} Let $K$ be an algebraically closed field and let $$ C\subseteq\mathbb P_K^2 $$ be a projective plane curve of degree $d$. Then there is a surjective morphism $$ C\longrightarrow\mathbb P_K^1 $$ such that each fibre consists of at most $d$ points. #### Proof {#br-ak-2012-l29-thm-01-proof} Choose a point $$ P\in\mathbb P_K^2 $$ not lying on the curve. Such a point exists because, in particular, $K$ is infinite. Consider projection away from $P$; its restriction induces a morphism $$ C\hookrightarrow\mathbb P_K^2\setminus\{P\} \longrightarrow\mathbb P_K^1. $$ The fibre of this morphism over a point $$ Q\in\mathbb P_K^1, $$ representing a direction at $P\in\mathbb P_K^2$ consists exactly of those points of the curve lying on the line determined by $Q$, $$ G=V_+(aX+bY+cZ)\cong\mathbb P_K^1\subseteq\mathbb P_K^2. $$ Thus the fibre over $Q$ can be described on $G$ by eliminating a variable from the curve equation $$ C=V_+(F) $$ using the line equation. The result is a nonzero homogeneous polynomial $\overline F$ in two variables of degree $d$; it cannot be zero, since then $P$ would lie on the curve. As we work over an algebraically closed field, $\overline F$ has at least one and at most $d$ zeros, all distinct from $P$. This proves surjectivity and the bound on the number of points in each fibre. $\square$ ### Theorem 29.3: a rational function as a morphism to $\mathbb P_K^1$ {#br-ak-2012-l29-thm-02} Let $K$ be a field and let $$ C\subseteq\mathbb P_K^2 $$ be a smooth irreducible projective plane curve. Let $$ D=C\cap D_+(Z)\cong K\!-\!\operatorname{Spek}(R) $$ be an affine piece of this curve, and let $$ q=\frac gh\in Q(R) $$ be a rational function, with $g,h\in R$ and $h\ne0$. Then there is exactly one morphism $$ \varphi:C\longrightarrow\mathbb P_K^1 $$ such that the diagram $$ \begin{matrix} D(h)&\stackrel{g/h}{\longrightarrow}&\mathbb A_K^1\cong D_+(s)\\ \downarrow&&\downarrow\\ C&\stackrel{\varphi}{\longrightarrow}&\mathbb P_K^1 \end{matrix} $$ commutes. Moreover, every genuine pole $P\in D$, meaning a point with $h(P)=0$ and $g(P)\ne0$, is mapped to the point at infinity $\infty\in\mathbb P_K^1$. #### Proof {#br-ak-2012-l29-thm-02-proof} First we define on $D$ an extension $$ \varphi:D\longrightarrow\mathbb P_K^1 $$ of the rational function $g/h$. If $q=0$, the constant map with value $0$ is the required extension, so assume from now on that $q\ne0$. Take a point $P\in D$ on the curve. If $P\in D(h)$, there is nothing to do. Thus suppose $h(P)=0$. > **Source correction REVIEW-AK-26-30-C12 - the zero rational function.** The source immediately writes $g/h=u\pi^n$ with $u$ a unit, which is possible only for a nonzero element of the function field. The theorem also permits $q=0$; the separate constant-map case above closes that gap without changing the nonzero case. Since the curve is smooth, Theorem 23.6 shows that its local ring $B$ at $P$ is a discrete valuation ring. The quotient $g/h$ can therefore be written there as $$ \frac gh=u\pi^n, $$ with $u\in B^\times$, $n\in\mathbb Z$, and $\pi$ a uniformiser (a generator of the maximal ideal). There is an open neighbourhood $$ P\in D(\psi)\subseteq D $$ such that $\pi$ and $u$ are defined on $D(\psi)$ and $u$ is a unit there. If $n\geq0$, then $$ \frac gh\in R_\psi, $$ so the point of indeterminacy is removable even for a map to $\mathbb A_K^1$. If $n\leq0$, the reciprocal quotient $$ \frac hg=u^{-1}\pi^{-n} $$ is defined on $D(\psi)$ as a map to $\mathbb A_K^1$. Using the “embedding with coordinates interchanged” $$ \mathbb A_K^1\cong D_+(t)\hookrightarrow\mathbb P_K^1, $$ we obtain a map to $\mathbb P_K^1$. We must show that these two morphisms to the projective line agree wherever both are defined. These are the points $P$ where $g/h$ has neither a zero nor a pole. Compatibility follows because on an open neighbourhood $$ P\in U $$ there is a map $$ \frac gh:U\longrightarrow (\mathbb A_K^1)^\times=\mathbb A_K^1\setminus\{0\}, $$ and the diagram $$ \begin{matrix} (\mathbb A_K^1)^\times&\stackrel{i^{-1}}{\longrightarrow}& \mathbb A_K^1\cong D_+(t)\\ \downarrow&&\downarrow\\ \mathbb A_K^1\cong D_+(s)&\longrightarrow&\mathbb P_K^1 \end{matrix} $$ commutes. This gives a well-defined morphism on the affine piece $D$. > **Source correction AGC-CORR-0127 - the object with zeros and poles.** The source refers to $\varphi$ in this sentence, although zeros and poles are properties of the rational function $g/h$. This edition displays the correct object without changing the overlap condition or the gluing diagram. For an arbitrary point on the projective curve $C$ and an affine neighbourhood $$ P\in D'\subseteq C, $$ we are in the same situation, because $$ D_+(h)\cap D'\ne\varnothing, $$ so the rational function is defined on a nonempty open set, possibly with different numerator and denominator. The preceding argument therefore applies in the same way. Uniqueness follows because, on every affine open set $$ P\in U $$ the intersection $U\cap D_+(h)$ is nonempty. A morphism from an integral variety to the affine line $\mathbb A_K^1$ is uniquely determined by its rational function. $\square$ ### Example 29.4: inversion on the projective line {#br-ak-2012-l29-ex-01} The inversion map $$ \begin{aligned} \mathbb A_K^1\supset D(z)&\longrightarrow\mathbb A_K^1,\\ z&\longmapsto z^{-1} \end{aligned} $$ extends to a bijective morphism $$ \begin{aligned} \mathbb P_K^1&\longrightarrow\mathbb P_K^1,\\ (x,y)&\longmapsto(y,x). \end{aligned} $$ This follows directly from Theorem 29.3. Every point $z\ne0$ is sent to $1/z$, while zero is sent to the point at infinity $\infty$. ## Parametrised projective plane curves {#br-ak-2012-l29-s03} Suppose a curve with rational parametrisation $$ s\longmapsto \left(\frac{\varphi_1(s)}{\psi(s)}, \frac{\varphi_2(s)}{\psi(s)}\right). $$ is given. In Theorem 6.11 we saw that its image satisfies an algebraic equation. In that theorem's proof we already used the homogenised parametrisation; it now reappears as a projective extension. ### Theorem 29.5: projective extension of a rational parametrisation {#br-ak-2012-l29-thm-03} Let $K$ be an infinite field, and let $$ \begin{aligned} \mathbb A_K^1\supseteq D(\psi)&\longrightarrow\mathbb A_K^2,\\ s&\longmapsto \left(\frac{\varphi_1(s)}{\psi(s)}, \frac{\varphi_2(s)}{\psi(s)}\right) \end{aligned} $$ be a rational parametrisation in reduced form, meaning that $\varphi_1,\varphi_2,\psi$ have no common divisor. Let $d$ be the largest degree of the polynomials involved, and let $$ \widehat{\varphi_1},\quad\widehat{\varphi_2},\quad\widehat\psi $$ be their homogenisations with respect to the new variable $t$. Each of $H_1,H_2,H_3$ is obtained from the corresponding homogenisation by multiplication by a suitable power of $t$, so that all three have degree $d$. Then $H_1,H_2,H_3$ define a morphism $$ \begin{aligned} H:\mathbb P_K^1&\longrightarrow\mathbb P_K^2,\\ (s,t)&\longmapsto\bigl(H_1(s,t),H_2(s,t),H_3(s,t)\bigr), \end{aligned} $$ such that the diagram $$ \begin{matrix} \mathbb A_K^1\supseteq D(\psi)&\longrightarrow& \mathbb A_K^2\cong D_+(Z)\\ \downarrow&&\downarrow\\ \mathbb P_K^1&\stackrel{H}{\longrightarrow}&\mathbb P_K^2 \end{matrix} $$ commutes. Moreover, the image of $H$ lies on the projective closure of the affine image curve. > **Source-condition note REVIEW-AK-26-30-C13 - infinitude of the base field.** The source does not state this hypothesis, but its proof uses the assertion that a finite open subset of $\mathbb P_K^1$ is empty. For the source's topology on $K$-points this requires $K$ to be infinite; it fails over a finite field, where every subset of $\mathbb P_K^1$ is finite. #### Proof {#br-ak-2012-l29-thm-03-proof} By Exercise 29.6, the map $H$ is well defined on all of $\mathbb P_K^1$, since $\varphi_1,\varphi_2,\psi$ have no common divisor. For commutativity, it suffices to note that a point $$ s\in D(\psi)\subseteq\mathbb A_K^1 $$ is sent, on the one hand, through $(s,1)$ to $$ \bigl(H_1(s,1),H_2(s,1),H_3(s,1)\bigr) =\bigl(\varphi_1(s),\varphi_2(s),\psi(s)\bigr), $$ and, on the other hand, to $$ \left(\frac{\varphi_1(s)}{\psi(s)}, \frac{\varphi_2(s)}{\psi(s)},1\right) =\bigl(\varphi_1(s),\varphi_2(s),\psi(s)\bigr) $$ as a projective point. For the additional assertion, let $C$ be the affine closure of the image and let $$ \overline C\subseteq\mathbb P_K^2 $$ be its projective closure. Consider the open complement $$ U=\mathbb P_K^2\setminus\overline C. $$ Since the map is continuous, the inverse image $H^{-1}(U)$ is open in $\mathbb P_K^1$ and can contain only points of $\mathbb P_K^1\setminus D(\psi)$. But a finite open subset of the projective line must be empty. $\square$ ### Theorem 29.6: projective closure of the graph of a polynomial {#br-ak-2012-l29-thm-04} Let $K$ be an algebraically closed field and let $$ F\in K[X] $$ be a polynomial in one variable of degree $d\geq1$. The projective closure $C$ of the graph $$ V(Y-F(X)) $$ is described by $$ V_+\bigl(YZ^{d-1}-\widehat F(X,Z)\bigr), $$ where $\widehat F(X,Z)$ is the homogenisation of $F$. If $d=1$ and $F=aX+b$, then $C$ has one additional point, the smooth point $(1,a,0)$. If $d\geq2$, the additional point is $(0,1,0)$, which is singular when $d\geq3$. For $d\geq2$, this point at infinity has multiplicity $d-1$. > **Source correction AGC-CORR-0128 - projective zero-locus operator.** The source prints $V$ for a homogeneous equation in $\mathbb P_K^2$. This edition uses $V_+$, in keeping with the projective ambient space and the notation of the subsequent proof. #### Proof {#br-ak-2012-l29-thm-04-proof} The equation of the projective closure follows directly from Corollary 28.10. To determine the intersection of $C$ with the projective line $V_+(Z)$ at infinity, set $Z=0$ in the equation. If $d=1$, the curve equation is the line equation $$ V_+(Y-aX-bZ), $$ and its intersection with $V_+(Z)$ gives the unique point $(1,a,0)$. If $d\geq2$, the curve equation is $$ V_+\bigl( YZ^{d-1}-s_dX^d-s_{d-1}X^{d-1}Z-\cdots-s_0Z^d \bigr), $$ with $s_d\ne0$. Setting $Z=0$ leaves $$ V_+(-s_dX^d), $$ so $X=0$. This gives the unique point at infinity $(0,1,0)$. To compute the multiplicity, consider the affine equation of the curve on $D_+(Y)$. Setting $Y=1$ gives the affine equation $$ V\bigl( Z^{d-1}-s_dX^d-s_{d-1}X^{d-1}Z-\cdots-s_0Z^d \bigr), $$ and in these coordinates $(0,1,0)$ becomes the origin. Hence its multiplicity is $d-1$, with the unique tangent line given by $Z=0$. If $d\geq3$, the multiplicity is at least $2$, so the point is singular. $\square$ > **Source correction AGC-U29-SRC-001 / AGC-CORR-0126 - undefined symbol in the singularity bound.** In the last sentence of the proof, the source prints $g\geq3$, although no quantity $g$ is defined in this theorem. The degree defined and used throughout the calculation is $d$; this edition therefore displays the intended bound as $d\geq3$ and explicitly retains the multiplicity argument $d-1\geq2$. The theorem can be understood as follows. If $d\geq2$, the point $(0,1,0)$ is the unique point at infinity and represents the direction of the $y$-axis. The line at infinity $V_+(Z)$ is the unique tangent line at this point. > **Source correction REVIEW-AK-26-30-C14 - direction versus asymptote.** The source calls the $y$-axis the graph's only asymptote. The projective point $(0,1,0)$ records the direction of that axis, not an affine asymptotic line; in the usual affine sense a polynomial graph of degree at least two has no linear asymptote. The source's valid tangent-line statement is retained. The normalisation of $C$ is $\mathbb P_K^1$. By Theorem 29.5, applied to the affine parametrisation of the graph $$ \begin{aligned} \mathbb A_K^1&\longrightarrow \mathbb A_K^1\times\mathbb A_K^1 =\mathbb A_K^2\cong D_+(Z)\subset\mathbb P_K^2,\\ x&\longmapsto(x,F(x))=(x,F(x),1), \end{aligned} $$ the normalisation map is given by $$ \begin{aligned} \mathbb P_K^1&\longrightarrow C\subset\mathbb P_K^2,\\ (x,t)&\longmapsto\bigl(xt^{d-1},\widehat F(x,t),t^d\bigr). \end{aligned} $$ The point at infinity $(1,0)$ is sent to $$ (0,s_d,0)=(0,1,0). $$ ### Theorem 29.7: projective closure of the graph of a rational function {#br-ak-2012-l29-thm-05} Let $K$ be an algebraically closed field, and let $$ G,H\in K[X] $$ be polynomials in one variable of degrees $d,e\geq1$, respectively, with no common root. Let $H\ne0$ and let $$ F(X)=\frac{G(X)}{H(X)} $$ be the corresponding rational function. Let $\widehat G(X,Z)$ and $\widehat H(X,Z)$ be their respective homogenisations. If $d>e$, the projective closure $C$ of the graph of $F(X)$ is described by $$ V_+\bigl(\widehat H(X,Z)YZ^{d-e-1}-\widehat G(X,Z)\bigr), $$ whereas if $d\leq e$, it is described by $$ V_+\bigl(\widehat H(X,Z)Y-\widehat G(X,Z)Z^{e-d+1}\bigr). $$ #### Proof {#br-ak-2012-l29-thm-05-proof} The affine description of the curve is $$ V(YH-G). $$ By Corollary 28.10, the projective closure is described by the homogenisation of $YH-G$. This is determined by the larger of the degrees of $YH$ and $G$; the summand of smaller degree must be “filled out” with a suitable power of $Z$. This yields the two equations above. $\square$ ## Monomial projective curves {#br-ak-2012-l29-s04} For the monomial plane curve $$ s\longmapsto(s^e,s^d)=(x,y) $$ with coprime exponents $e>d$, Theorem 29.5 gives the monomial projective curve $$ (s,t)\longmapsto(s^e,s^dt^{e-d},t^e). $$ On the open set $D_+(t)$ this is the original map, whereas on $D_+(s)$ it becomes the affine map $$ t\longmapsto(t^{e-d},t^e). $$ ### Theorem 29.8: singularities of monomial projective curves {#br-ak-2012-l29-thm-06} Let $e>d$ be coprime. For the monomial projective plane curve of degree $e$ $$ C:\quad(s,t)\longmapsto\bigl(s^e,s^dt^{e-d},t^e\bigr), $$ the following statements hold. 1. The curve is described by the homogeneous equation of degree $e$ $$ Y^e=X^dZ^{e-d}. $$ 2. The curve is smooth at all points other than $(0,0,1)$ and $(1,0,0)$. 3. The curve has multiplicity $d$ at $(0,0,1)$ and multiplicity $e-d$ at $(1,0,0)$. 4. If $e\geq3$, the curve is not smooth. #### Proof {#br-ak-2012-l29-thm-06-proof} 1. The affine equation is $X^d-Y^e$. By Corollary 28.10, the projective closure is described by its homogenisation, namely $$ V_+\bigl(X^dZ^{e-d}-Y^e\bigr). $$ > **Source correction AGC-CORR-0129 - projective zero-locus operator.** The source prints $V$ for the homogenisation defining the closure in $\mathbb P_K^2$; this edition uses $V_+$. The affine loci below retain $V$. 2. On the affine curve $$ V(X^d-Y^e)\subseteq\mathbb A_K^2\subseteq\mathbb P_K^2, $$ by the current source's [normalisation result for affine monomial curves](https://de.wikiversity.org/wiki/Affine_Kurven/Monomiale_Kurvenabbildung/Ist_Normalisierung/Fakt), only the origin—corresponding to the projective point $(0,0,1)$—can fail to be smooth. Points of the curve outside $D_+(Z)$ are obtained by setting $Z=0$ in the equation. This forces $Y=0$, leaving only the point $(1,0,0)$. > **Semantic-source note REVIEW-AK-26-30-C15.** The expanded 2012 witness cites Theorem 20.12 here; the frozen current proof instead links to the named normalisation result above. This edition preserves the current source target and the unchanged smoothness conclusion. 3. Multiplicity at a point is a local property. The point $(0,0,1)$ corresponds to the origin on the affine monomial curve $$ V(X^d-Y^e), $$ which, by Corollary 23.8, has multiplicity equal to the smaller exponent, namely $d$. The point $(1,0,0)$ lies in $D_+(X)$, where the affine equation is $$ V(Y^e-Z^{e-d}). $$ Its multiplicity is again the smaller exponent, namely $e-d$. 4. This follows from item 3. $\square$