Algebraic Geometry

# Algebraic Geometry

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Paper 1, Section II, I

commentLet $k$ be an algebraically closed field and let $V \subset \mathbb{A}_{k}^{n}$ be a non-empty affine variety. Show that $V$ is a finite union of irreducible subvarieties.

Let $V_{1}$ and $V_{2}$ be subvarieties of $\mathbb{A}_{k}^{n}$ given by the vanishing loci of ideals $I_{1}$ and $I_{2}$ respectively. Prove the following assertions.

(i) The variety $V_{1} \cap V_{2}$ is equal to the vanishing locus of the ideal $I_{1}+I_{2}$.

(ii) The variety $V_{1} \cup V_{2}$ is equal to the vanishing locus of the ideal $I_{1} \cap I_{2}$.

Decompose the vanishing locus

$\mathbb{V}\left(X^{2}+Y^{2}-1, X^{2}-Z^{2}-1\right) \subset \mathbb{A}_{\mathbb{C}}^{3}$

into irreducible components.

Let $V \subset \mathbb{A}_{k}^{3}$ be the union of the three coordinate axes. Let $W$ be the union of three distinct lines through the point $(0,0)$ in $\mathbb{A}_{k}^{2}$. Prove that $W$ is not isomorphic to $V$.

Paper 2, Section II, I

commentLet $k$ be an algebraically closed field and $n \geqslant 1$. Exhibit $G L(n, k)$ as an open subset of affine space $\mathbb{A}_{k}^{n^{2}}$. Deduce that $G L(n, k)$ is smooth. Prove that it is also irreducible.

Prove that $G L(n, k)$ is isomorphic to a closed subvariety in an affine space.

Show that the matrix multiplication map

$G L(n, k) \times G L(n, k) \rightarrow G L(n, k)$

that sends a pair of matrices to their product is a morphism.

Prove that any morphism from $\mathbb{A}_{k}^{n}$ to $\mathbb{A}_{k}^{1}\setminus\{0\}$ is constant.

Prove that for $n \geqslant 2$ any morphism from $\mathbb{P}_{k}^{n}$ to $\mathbb{P}_{k}^{1}$ is constant.

Paper 3, Section II, I

commentIn this question, all varieties are over an algebraically closed field $k$ of characteristic zero.

What does it mean for a projective variety to be smooth? Give an example of a smooth affine variety $X \subset \mathbb{A}_{k}^{n}$ whose projective closure $\bar{X} \subset \mathbb{P}_{k}^{n}$ is not smooth.

What is the genus of a smooth projective curve? Let $X \subset \mathbb{P}_{k}^{4}$ be the hypersurface $V\left(X_{0}^{3}+X_{1}^{3}+X_{2}^{3}+X_{3}^{3}+X_{4}^{3}\right)$. Prove that $X$ contains a smooth curve of genus $1 .$

Let $C \subset \mathbb{P}_{k}^{2}$ be an irreducible curve of degree 2 . Prove that $C$ is isomorphic to $\mathbb{P}_{k}^{1}$.

We define a generalized conic in $\mathbb{P}_{k}^{2}$ to be the vanishing locus of a non-zero homogeneous quadratic polynomial in 3 variables. Show that there is a bijection between the set of generalized conics in $\mathbb{P}_{k}^{2}$ and the projective space $\mathbb{P}_{k}^{5}$, which maps the conic $V(f)$ to the point whose coordinates are the coefficients of $f$.

(i) Let $R^{\circ} \subset \mathbb{P}_{k}^{5}$ be the subset of conics that consist of unions of two distinct lines. Prove that $R^{\circ}$ is not Zariski closed, and calculate its dimension.

(ii) Let $I$ be the homogeneous ideal of polynomials vanishing on $R^{\circ}$. Determine generators for the ideal $I$.

Paper 4, Section II, I

commentLet $C$ be a smooth irreducible projective algebraic curve over an algebraically closed field.

Let $D$ be an effective divisor on $C$. Prove that the vector space $L(D)$ of rational functions with poles bounded by $D$ is finite dimensional.

Let $D$ and $E$ be linearly equivalent divisors on $C$. Exhibit an isomorphism between the vector spaces $L(D)$ and $L(E)$.

What is a canonical divisor on $C$ ? State the Riemann-Roch theorem and use it to calculate the degree of a canonical divisor in terms of the genus of $C$.

Prove that the canonical divisor on a smooth cubic plane curve is linearly equivalent to the zero divisor.

Paper 1, Section II, F

commentLet $k$ be an algebraically closed field of characteristic zero. Prove that an affine variety $V \subset \mathbb{A}_{k}^{n}$ is irreducible if and only if the associated ideal $I(V)$ of polynomials that vanish on $V$ is prime.

Prove that the variety $\mathbb{V}\left(y^{2}-x^{3}\right) \subset \mathbb{A}_{k}^{2}$ is irreducible.

State what it means for an affine variety over $k$ to be smooth and determine whether or not $\mathbb{V}\left(y^{2}-x^{3}\right)$ is smooth.

Paper 2, Section II, F

commentLet $k$ be an algebraically closed field of characteristic not equal to 2 and let $V \subset \mathbb{P}_{k}^{3}$ be a nonsingular quadric surface.

(a) Prove that $V$ is birational to $\mathbb{P}_{k}^{2}$.

(b) Prove that there exists a pair of disjoint lines on $V$.

(c) Prove that the affine variety $W=\mathbb{V}(x y z-1) \subset \mathbb{A}_{k}^{3}$ does not contain any lines.

Paper 3, Section II, F

comment(i) Suppose $f(x, y)=0$ is an affine equation whose projective completion is a smooth projective curve. Give a basis for the vector space of holomorphic differential forms on this curve. [You are not required to prove your assertion.]

Let $C \subset \mathbb{P}^{2}$ be the plane curve given by the vanishing of the polynomial

$X_{0}^{4}-X_{1}^{4}-X_{2}^{4}=0$

over the complex numbers.

(ii) Prove that $C$ is nonsingular.

(iii) Let $\ell$ be a line in $\mathbb{P}^{2}$ and define $D$ to be the divisor $\ell \cap C$. Prove that $D$ is a canonical divisor on $C$.

(iv) Calculate the minimum degree $d$ such that there exists a non-constant map

$C \rightarrow \mathbb{P}^{1}$

of degree $d$.

[You may use any results from the lectures provided that they are stated clearly.]

Paper 4, Section II, F

commentLet $P_{0}, \ldots, P_{n}$ be a basis for the homogeneous polynomials of degree $n$ in variables $Z_{0}$ and $Z_{1}$. Then the image of the $\operatorname{map} \mathbb{P}^{1} \rightarrow \mathbb{P}^{n}$ given by

$\left[Z_{0}, Z_{1}\right] \mapsto\left[P_{0}\left(Z_{0}, Z_{1}\right), \ldots, P_{n}\left(Z_{0}, Z_{1}\right)\right]$

is called a rational normal curve.

Let $p_{1}, \ldots, p_{n+3}$ be a collection of points in general linear position in $\mathbb{P}^{n}$. Prove that there exists a unique rational normal curve in $\mathbb{P}^{n}$ passing through these points.

Choose a basis of homogeneous polynomials of degree 3 as above, and give generators for the homogeneous ideal of the corresponding rational normal curve.

Paper 1, Section II, F

comment(a) Let $k$ be an algebraically closed field of characteristic 0 . Consider the algebraic variety $V \subset \mathbb{A}^{3}$ defined over $k$ by the polynomials

$x y, \quad y^{2}-z^{3}+x z, \quad \text { and } x(x+y+2 z+1)$

Determine

(i) the irreducible components of $V$,

(ii) the tangent space at each point of $V$,

(iii) for each irreducible component, the smooth points of that component, and

(iv) the dimensions of the irreducible components.

(b) Let $L \supseteq K$ be a finite extension of fields, and $\operatorname{dim}_{K} L=n$. Identify $L$ with $\mathbb{A}^{n}$ over $K$ and show that

$U=\{\alpha \in L \mid K[\alpha]=L\}$

is the complement in $\mathbb{A}^{n}$ of the vanishing set of some polynomial. [You need not show that $U$ is non-empty. You may assume that $K[\alpha]=L$ if and only if $1, \alpha, \ldots, \alpha^{n-1}$ form a basis of $L$ over $K$.]

Paper 2, Section II, F

comment(a) Let $A$ be a commutative algebra over a field $k$, and $p: A \rightarrow k$ a $k$-linear homomorphism. Define $\operatorname{Der}(A, p)$, the derivations of $A$ centered in $p$, and define the tangent space $T_{p} A$ in terms of this.

Show directly from your definition that if $f \in A$ is not a zero divisor and $p(f) \neq 0$, then the natural map $T_{p} A\left[\frac{1}{f}\right] \rightarrow T_{p} A$ is an isomorphism.

(b) Suppose $k$ is an algebraically closed field and $\lambda_{i} \in k$ for $1 \leqslant i \leqslant r$. Let

$X=\left\{(x, y) \in \mathbb{A}^{2} \mid x \neq 0, y \neq 0, y^{2}=\left(x-\lambda_{1}\right) \cdots\left(x-\lambda_{r}\right)\right\}$

Find a surjective map $X \rightarrow \mathbb{A}^{1}$. Justify your answer.

Paper 3, Section II, F

commentLet $W \subseteq \mathbb{A}^{2}$ be the curve defined by the equation $y^{3}=x^{4}+1$ over the complex numbers $\mathbb{C}$, and let $X \subseteq \mathbb{P}^{2}$ be its closure.

(a) Show $X$ is smooth.

(b) Determine the ramification points of the $\operatorname{map} X \rightarrow \mathbb{P}^{1}$ defined by

$(x: y: z) \mapsto(x: z) .$

Using this, determine the Euler characteristic and genus of $X$, stating clearly any theorems that you are using.

(c) Let $\omega=\frac{d x}{y^{2}} \in \mathcal{K}_{X}$. Compute $\nu_{p}(\omega)$ for all $p \in X$, and determine a basis for $\mathcal{L}\left(\mathcal{K}_{X}\right)$

Paper 4, Section II, F

comment(a) Let $X \subseteq \mathbb{P}^{2}$ be a smooth projective plane curve, defined by a homogeneous polynomial $F(x, y, z)$ of degree $d$ over the complex numbers $\mathbb{C}$.

(i) Define the divisor $[X \cap H]$, where $H$ is a hyperplane in $\mathbb{P}^{2}$ not contained in $X$, and prove that it has degree $d$.

(ii) Give (without proof) an expression for the degree of $\mathcal{K}_{X}$ in terms of $d$.

(iii) Show that $X$ does not have genus 2 .

(b) Let $X$ be a smooth projective curve of genus $g$ over the complex numbers $\mathbb{C}$. For $p \in X$ let

$G(p)=\left\{n \in \mathbb{N} \mid\right.$ there is no $f \in k(X)$ with $v_{p}(f)=n$, and $v_{q}(f) \leqslant 0$ for all $\left.q \neq p\right\} .$

(i) Define $\ell(D)$, for a divisor $D$.

(ii) Show that for all $p \in X$,

$\ell(n p)= \begin{cases}\ell((n-1) p) & \text { for } n \in G(p) \\ \ell((n-1) p)+1 & \text { otherwise }\end{cases}$

(iii) Show that $G(p)$ has exactly $g$ elements. [Hint: What happens for large $n$ ?]

(iv) Now suppose that $X$ has genus 2 . Show that $G(p)=\{1,2\}$ or $G(p)=\{1,3\}$.

[In this question $\mathbb{N}$ denotes the set of positive integers.]

Paper 1, Section II, I

comment(a) Let $k$ be an uncountable field, $\mathcal{M} \subseteq k\left[x_{1}, \ldots, x_{n}\right]$ a maximal ideal and $A=k\left[x_{1}, \ldots, x_{n}\right] / \mathcal{M} .$

Show that every element of $A$ is algebraic over $k$.

(b) Now assume that $k$ is algebraically closed. Suppose that $J \subset k\left[x_{1}, \ldots, x_{n}\right]$ is an ideal, and that $f \in k\left[x_{1}, \ldots, x_{n}\right]$ vanishes on $Z(J)$. Using the result of part (a) or otherwise, show that $f^{N} \in J$ for some $N \geqslant 1$.

(c) Let $f: X \rightarrow Y$ be a morphism of affine algebraic varieties. Show $\overline{f(X)}=Y$ if and only if the map $f^{*}: k[Y] \rightarrow k[X]$ is injective.

Suppose now that $\overline{f(X)}=Y$, and that $X$ and $Y$ are irreducible. Define the dimension of $X, \operatorname{dim} X$, and show $\operatorname{dim} X \geqslant \operatorname{dim} Y$. [You may use whichever definition of $\operatorname{dim} X$ you find most convenient.]

Paper 2, Section II, I

comment(a) Let $X \subseteq \mathbf{A}^{n}$ be an affine algebraic variety defined over the field $k$.

Define the tangent space $T_{p} X$ for $p \in X$, and the dimension of $X$ in terms of $T_{p} X$.

Suppose that $k$ is an algebraically closed field with char $k>0$. Show directly from your definition that if $X=Z(f)$, where $f \in k\left[x_{1}, \ldots, x_{n}\right]$ is irreducible, then $\operatorname{dim} X=n-1$.

[Any form of the Nullstellensatz may be used if you state it clearly.]

(b) Suppose that char $k=0$, and let $W$ be the vector space of homogeneous polynomials of degree $d$ in 3 variables over $k$. Show that

$U=\left\{(f, p) \in W \times k^{3} \mid Z(f-1) \text { is a smooth surface at } p\right\}$

is a non-empty Zariski open subset of $W \times k^{3}$.

Paper 3, Section II, I

comment(a) State the Riemann-Roch theorem.

(b) Let $E$ be a smooth projective curve of genus 1 over an algebraically closed field $k$, with char $k \neq 2,3$. Show that there exists an isomorphism from $E$ to the plane cubic in $\mathbf{P}^{2}$ defined by the equation

$y^{2}=\left(x-\lambda_{1}\right)\left(x-\lambda_{2}\right)\left(x-\lambda_{3}\right)$

for some distinct $\lambda_{1}, \lambda_{2}, \lambda_{3} \in k$.

(c) Let $Q$ be the point at infinity on $E$. Show that the map $E \rightarrow C l^{0}(E), P \mapsto[P-Q]$ is an isomorphism.

Describe how this defines a group structure on $E$. Denote addition by $\boxplus$. Determine all the points $P \in E$ with $P \boxplus P=Q$ in terms of the equation of the plane curve in part (b).

Paper 4, Section II, I

commentState a theorem which describes the canonical divisor of a smooth plane curve $C$ in terms of the divisor of a hyperplane section. Express the degree of the canonical divisor $K_{C}$ and the genus of $C$ in terms of the degree of $C$. [You need not prove these statements.]

From now on, we work over $\mathbb{C}$. Consider the curve in $\mathbf{A}^{2}$ defined by the equation

$y+x^{3}+x y^{3}=0$

Let $C$ be its projective completion. Show that $C$ is smooth.

Compute the genus of $C$ by applying the Riemann-Hurwitz theorem to the morphism $C \rightarrow \mathbf{P}^{1}$ induced from the rational map $(x, y) \mapsto y$. [You may assume that the discriminant of $x^{3}+a x+b$ is $-4 a^{3}-27 b^{2}$.]

Paper 1, Section II, I

commentLet $k$ be an algebraically closed field.

(a) Let $X$ and $Y$ be varieties defined over $k$. Given a function $f: X \rightarrow Y$, define what it means for $f$ to be a morphism of varieties.

(b) If $X$ is an affine variety, show that the coordinate ring $A(X)$ coincides with the ring of regular functions on $X$. [Hint: You may assume a form of the Hilbert Nullstellensatz.]

(c) Now suppose $X$ and $Y$ are affine varieties. Show that if $X$ and $Y$ are isomorphic, then there is an isomorphism of $k$-algebras $A(X) \cong A(Y)$.

(d) Show that $Z\left(x^{2}-y^{3}\right) \subseteq \mathbb{A}^{2}$ is not isomorphic to $\mathbb{A}^{1}$.

Paper 2, Section II, I

commentLet $k$ be an algebraically closed field of any characteristic.

(a) Define what it means for a variety $X$ to be non-singular at a point $P \in X$.

(b) Let $X \subseteq \mathbb{P}^{n}$ be a hypersurface $Z(f)$ for $f \in k\left[x_{0}, \ldots, x_{n}\right]$ an irreducible homogeneous polynomial. Show that the set of singular points of $X$ is $Z(I)$, where $I \subseteq$ $k\left[x_{0}, \ldots, x_{n}\right]$ is the ideal generated by $\partial f / \partial x_{0}, \ldots, \partial f / \partial x_{n} .$

(c) Consider the projective plane curve corresponding to the affine curve in $\mathbb{A}^{2}$ given by the equation

$x^{4}+x^{2} y^{2}+y^{2}+1=0 .$

Find the singular points of this projective curve if char $k \neq 2$. What goes wrong if char $k=2$ ?

Paper 3, Section II, I

comment(a) Define what it means to give a rational map between algebraic varieties. Define a birational map.

(b) Let

$X=Z\left(y^{2}-x^{2}(x-1)\right) \subseteq \mathbb{A}^{2}$

Define a birational map from $X$ to $\mathbb{A}^{1}$. [Hint: Consider lines through the origin.]

(c) Let $Y \subseteq \mathbb{A}^{3}$ be the surface given by the equation

$x_{1}^{2} x_{2}+x_{2}^{2} x_{3}+x_{3}^{2} x_{1}=0 .$

Consider the blow-up $X \subseteq \mathbb{A}^{3} \times \mathbb{P}^{2}$ of $\mathbb{A}^{3}$ at the origin, i.e. the subvariety of $\mathbb{A}^{3} \times \mathbb{P}^{2}$ defined by the equations $x_{i} y_{j}=x_{j} y_{i}$ for $1 \leqslant i<j \leqslant 3$, with $y_{1}, y_{2}, y_{3}$ coordinates on $\mathbb{P}^{2}$. Let $\varphi: X \rightarrow \mathbb{A}^{3}$ be the projection and $E=\varphi^{-1}(0)$. Recall that the proper transform $\tilde{Y}$ of $Y$ is the closure of $\varphi^{-1}(Y) \backslash E$ in $X$. Give equations for $\tilde{Y}$, and describe the fibres of the morphism $\left.\varphi\right|_{\widetilde{Y}}: \widetilde{Y} \rightarrow Y$.

Paper 4, Section II, I

comment(a) Let $X$ and $Y$ be non-singular projective curves over a field $k$ and let $\varphi: X \rightarrow Y$ be a non-constant morphism. Define the ramification degree $e_{P}$ of $\varphi$ at a point $P \in X$.

(b) Suppose char $k \neq 2$. Let $X=Z(f)$ be the plane cubic with $f=x_{0} x_{2}^{2}-x_{1}^{3}+x_{0}^{2} x_{1}$, and let $Y=\mathbb{P}^{1}$. Explain how the projection

$\left(x_{0}: x_{1}: x_{2}\right) \mapsto\left(x_{0}: x_{1}\right)$

defines a morphism $\varphi: X \rightarrow Y$. Determine the degree of $\varphi$ and the ramification degrees $e_{P}$ for all $P \in X$.

(c) Let $X$ be a non-singular projective curve and let $P \in X$. Show that there is a non-constant rational function on $X$ which is regular on $X \backslash\{P\}$.

Paper 1, Section II, H

commentLet $k$ be an algebraically closed field.

(a) Let $X$ and $Y$ be affine varieties defined over $k$. Given a map $f: X \rightarrow Y$, define what it means for $f$ to be a morphism of affine varieties.

(b) Let $f: \mathbb{A}^{1} \rightarrow \mathbb{A}^{3}$ be the map given by

$f(t)=\left(t, t^{2}, t^{3}\right)$

Show that $f$ is a morphism. Show that the image of $f$ is a closed subvariety of $\mathbb{A}^{3}$ and determine its ideal.

(c) Let $g: \mathbb{P}^{1} \times \mathbb{P}^{1} \times \mathbb{P}^{1} \rightarrow \mathbb{P}^{7}$ be the map given by

$g\left(\left(s_{1}, t_{1}\right),\left(s_{2}, t_{2}\right),\left(s_{3}, t_{3}\right)\right)=\left(s_{1} s_{2} s_{3}, s_{1} s_{2} t_{3}, s_{1} t_{2} s_{3}, s_{1} t_{2} t_{3}, t_{1} s_{2} s_{3}, t_{1} s_{2} t_{3}, t_{1} t_{2} s_{3}, t_{1} t_{2} t_{3}\right) .$

Show that the image of $g$ is a closed subvariety of $\mathbb{P}^{7}$.

Paper 2, Section II, H

commentIn this question we work over an algebraically closed field of characteristic zero. Let $X^{o}=Z\left(x^{6}+x y^{5}+y^{6}-1\right) \subset \mathbb{A}^{2}$ and let $X \subset \mathbb{P}^{2}$ be the closure of $X^{o}$ in $\mathbb{P}^{2} .$

(a) Show that $X$ is a non-singular curve.

(b) Show that $\omega=d x /\left(5 x y^{4}+6 y^{5}\right)$ is a regular differential on $X$.

(c) Compute the divisor of $\omega$. What is the genus of $X$ ?

Paper 3, Section II, H

comment(a) Let $X$ be an affine variety. Define the tangent space of $X$ at a point $P$. Say what it means for the variety to be singular at $P$.

Define the dimension of $X$ in terms of (i) the tangent spaces of $X$, and (ii) Krull dimension.

(b) Consider the ideal $I$ generated by the set $\left\{y, y^{2}-x^{3}+x y^{3}\right\} \subseteq k[x, y]$. What is $Z(I) \subseteq \mathbb{A}^{2} ?$

Using the generators of the ideal, calculate the tangent space of a point in $Z(I)$. What has gone wrong? [A complete argument is not necessary.]

(c) Calculate the dimension of the tangent space at each point $p \in X$ for $X=$ $Z\left(x-y^{2}, x-z w\right) \subseteq \mathbb{A}^{4}$, and determine the location of the singularities of $X .$

Paper 4, Section II, H

comment(a) Let $C$ be a smooth projective curve, and let $D$ be an effective divisor on $C$. Explain how $D$ defines a morphism $\phi_{D}$ from $C$ to some projective space.

State a necessary and sufficient condition on $D$ so that the pull-back of a hyperplane via $\phi_{D}$ is an element of the linear system $|D|$.

State necessary and sufficient conditions for $\phi_{D}$ to be an isomorphism onto its image.

(b) Let $C$ now have genus 2 , and let $K$ be an effective canonical divisor. Show that the morphism $\phi_{K}$ is a morphism of degree 2 from $C$ to $\mathbb{P}^{1}$.

Consider the divisor $K+P_{1}+P_{2}$ for points $P_{i}$ with $P_{1}+P_{2} \nsim K$. Show that the linear system associated to this divisor induces a morphism $\phi$ from $C$ to a quartic curve in $\mathbb{P}^{2}$. Show furthermore that $\phi(P)=\phi(Q)$, with $P \neq Q$, if and only if $\{P, Q\}=\left\{P_{1}, P_{2}\right\}$.

[You may assume the Riemann-Roch theorem.]

Paper 1, Section II, F

commentLet $k$ be an algebraically closed field.

(i) Let $X$ and $Y$ be affine varieties defined over $k$. Given a map $f: X \rightarrow Y$, define what it means for $f$ to be a morphism of affine varieties.

(ii) With $X, Y$ still affine varieties over $k$, show that there is a one-to-one correspondence between $\operatorname{Hom}(X, Y)$, the set of morphisms between $X$ and $Y$, and $\operatorname{Hom}(A(Y), A(X))$, the set of $k$-algebra homomorphisms between $A(Y)$ and $A(X)$.

(iii) Let $f: \mathbb{A}^{2} \rightarrow \mathbb{A}^{4}$ be given by $f(t, u)=\left(u, t, t^{2}, t u\right)$. Show that the image of $f$ is an affine variety $X$, and find a set of generators for $I(X)$.

Paper 2, Section II, F

comment(i) Define the radical of an ideal.

(ii) Assume the following statement: If $k$ is an algebraically closed field and $I \subseteq$ $k\left[x_{1}, \ldots, x_{n}\right]$ is an ideal, then either $I=(1)$ or $Z(I) \neq \emptyset$. Prove the Hilbert Nullstellensatz, namely that if $I \subseteq k\left[x_{1}, \ldots, x_{n}\right]$ with $k$ algebraically closed, then

$I(Z(I))=\sqrt{I}$

(iii) Show that if $A$ is a commutative ring and $I, J \subseteq A$ are ideals, then

$\sqrt{I \cap J}=\sqrt{I} \cap \sqrt{J} .$

(iv) Is

$\sqrt{I+J}=\sqrt{I}+\sqrt{J} ?$

Give a proof or a counterexample.

Paper 3, Section II, F

comment(i) Let $X$ be an affine variety. Define the tangent space of $X$ at a point $P$. Say what it means for the variety to be singular at $P$.

(ii) Find the singularities of the surface in $\mathbb{P}^{3}$ given by the equation

$x y z+y z w+z w x+w x y=0 .$

(iii) Consider $C=Z\left(x^{2}-y^{3}\right) \subseteq \mathbb{A}^{2}$. Let $X \rightarrow \mathbb{A}^{2}$ be the blowup of the origin. Compute the proper transform of $C$ in $X$, and show it is non-singular.

Paper 4, Section II, F

comment(i) Explain how a linear system on a curve $C$ may induce a morphism from $C$ to projective space. What condition on the linear system is necessary to yield a morphism $f: C \rightarrow \mathbb{P}^{n}$ such that the pull-back of a hyperplane section is an element of the linear system? What condition is necessary to imply the morphism is an embedding?

(ii) State the Riemann-Roch theorem for curves.

(iii) Show that any divisor of degree 5 on a curve $C$ of genus 2 induces an embedding.

Paper 1, Section II, H

commentLet $k$ be an algebraically closed field and $n \geqslant 1$. We say that $f \in k\left[x_{1}, \ldots, x_{n}\right]$ is singular at $p \in \mathbf{A}^{n}$ if either $p$ is a singularity of the hypersurface $\{f=0\}$ or $f$ has an irreducible factor $h$ of multiplicity strictly greater than one with $h(p)=0$. Given $d \geqslant 1$, let $X=\left\{f \in k\left[x_{1}, \ldots, x_{n}\right] \mid \operatorname{deg} f \leqslant d\right\}$ and let

$Y=\left\{(f, p) \in X \times \mathbf{A}^{n} \mid f \text { is singular at } p\right\}$

(i) Show that $X \simeq \mathbf{A}^{N}$ for some $N$ (you need not determine $N$ ) and that $Y$ is a Zariski closed subvariety of $X \times \mathbf{A}^{n}$.

(ii) Show that the fibres of the projection map $Y \rightarrow \mathbf{A}^{n}$ are linear subspaces of $\operatorname{dim} e n s i o n N-(n+1)$. Conclude that $\operatorname{dim} Y<\operatorname{dim} X$.

(iii) Hence show that $\{f \in X \mid \operatorname{deg} f=d, Z(f)$ smooth $\}$ is dense in $X$.

[You may use standard results from lectures if they are accurately quoted.]

Paper 2, Section II, H

comment(i) Let $k$ be an algebraically closed field, $n \geqslant 1$, and $S$ a subset of $k^{n}$.

Let $I(S)=\left\{f \in k\left[x_{1}, \ldots, x_{n}\right] \mid f(p)=0\right.$ when $\left.p \in S\right\}$. Show that $I(S)$ is an ideal, and that $k\left[x_{1}, \ldots, x_{n}\right] / I(S)$ does not have any non-zero nilpotent elements.

Let $X \subseteq \mathbf{A}^{n}, Y \subseteq \mathbf{A}^{m}$ be affine varieties, and $\Phi: k[Y] \rightarrow k[X]$ be a $k$-algebra homomorphism. Show that $\Phi$ determines a map of sets from $X$ to $Y$.

(ii) Let $X$ be an irreducible affine variety. Define the dimension of $X, \operatorname{dim} X$ (in terms of the tangent spaces of $X$ ) and the transcendence dimension of $X, \operatorname{tr} \cdot \operatorname{dim} X$.

State the Noether normalization theorem. Using this, or otherwise, prove that the transcendence dimension of $X$ equals the dimension of $X$.

Paper 3, Section II, H

commentLet $f \in k[x]$ be a polynomial with distinct roots, $\operatorname{deg} f=d>2$, char $k=0$, and let $C \subseteq \mathbf{P}^{2}$ be the projective closure of the affine curve

$y^{d-1}=f(x)$

Show that $C$ is smooth, with a single point at $\infty$.

Pick an appropriate $\omega \in \Omega_{k(C) / k}^{1}$ and compute the valuation $v_{q}(\omega)$ for all $q \in C$.

Hence determine $\operatorname{deg} \mathcal{K}_{C}$.

Paper 4, Section II, H

commentLet $X$ be a smooth projective curve of genus $g>0$ over an algebraically closed field of characteristic $\neq 2$, and suppose there is a degree 2 morphism $\pi: X \rightarrow \mathbf{P}^{1}$. How many ramification points of $\pi$ are there?

Suppose $Q$ and $R$ are distinct ramification points of $\pi$. Show that $Q \nsim R$, but $2 Q \sim 2 R$.

Now suppose $g=2$. Show that every divisor of degree 2 on $X$ is linearly equivalent to $P+P^{\prime}$ for some $P, P^{\prime} \in X$, and deduce that every divisor of degree 0 is linearly equivalent to $P_{1}-P_{2}$ for some $P_{1}, P_{2} \in X$.

Show that the subgroup $\left\{[D] \in C l^{0}(X) \mid 2[D]=0\right\}$ of the divisor class group of $X$ has order $16 .$

Paper 1, Section II, H

commentLet $V \subset \mathbb{A}^{n}$ be an affine variety over an algebraically closed field $k$. What does it mean to say that $V$ is irreducible? Show that any non-empty affine variety $V \subset \mathbb{A}^{n}$ is the union of a finite number of irreducible affine varieties $V_{j} \subset \mathbb{A}^{n}$.

Define the ideal $I(V)$ of $V$. Show that $I(V)$ is a prime ideal if and only if $V$ is irreducible.

Assume that the base field $k$ has characteristic zero. Determine the irreducible components of

$V\left(X_{1} X_{2}, X_{1} X_{3}+X_{2}^{2}-1, X_{1}^{2}\left(X_{1}-X_{3}\right)\right) \subset \mathbb{A}^{3}$

Paper 2, Section II, H

commentLet $V \subset \mathbb{P}^{3}$ be an irreducible quadric surface.

(i) Show that if $V$ is singular, then every nonsingular point lies in exactly one line in $V$, and that all the lines meet in the singular point, which is unique.

(ii) Show that if $V$ is nonsingular then each point of $V$ lies on exactly two lines of $V$.

Let $V$ be nonsingular, $P_{0}$ a point of $V$, and $\Pi \subset \mathbb{P}^{3}$ a plane not containing $P_{0}$. Show that the projection from $P_{0}$ to $\Pi$ is a birational map $f: V \rightarrow \rightarrow \Pi$. At what points does $f$ fail to be regular? At what points does $f^{-1}$ fail to be regular? Justify your answers.

Paper 3, Section II, H

commentLet $C \subset \mathbb{P}^{2}$ be the plane curve given by the polynomial

$X_{0}^{n}-X_{1}^{n}-X_{2}^{n}$

over the field of complex numbers, where $n \geqslant 3$.

(i) Show that $C$ is nonsingular.

(ii) Compute the divisors of the rational functions

$x=\frac{X_{1}}{X_{0}}, \quad y=\frac{X_{2}}{X_{0}}$

on $C$.

(iii) Consider the morphism $\phi=\left(X_{0}: X_{1}\right): C \rightarrow \mathbb{P}^{1}$. Compute its ramification points and degree.

(iv) Show that a basis for the space of regular differentials on $C$ is

$\left\{x^{i} y^{j} \omega_{0} \mid i, j \geqslant 0, i+j \leqslant n-3\right\}$

where $\omega_{0}=d x / y^{n-1} .$

Paper 4, Section II, H

commentLet $C$ be a nonsingular projective curve, and $D$ a divisor on $C$ of degree $d$.

(i) State the Riemann-Roch theorem for $D$, giving a brief explanation of each term. Deduce that if $d>2 g-2$ then $\ell(D)=1-g+d$.

(ii) Show that, for every $P \in C$,

$\ell(D-P) \geqslant \ell(D)-1$

Deduce that $\ell(D) \leqslant 1+d$. Show also that if $\ell(D)>1$, then $\ell(D-P)=\ell(D)-1$ for all but finitely many $P \in C$.

(iii) Deduce that for every $d \geqslant g-1$ there exists a divisor $D$ of degree $d$ with $\ell(D)=1-g+d$.

Paper 1, Section II, I

comment(a) Let $X$ be an affine variety, $k[X]$ its ring of functions, and let $p \in X$. Assume $k$ is algebraically closed. Define the tangent space $T_{p} X$ at $p$. Prove the following assertions.

(i) A morphism of affine varieties $f: X \rightarrow Y$ induces a linear map

$d f: T_{p} X \rightarrow T_{f(p)} Y$

(ii) If $g \in k[X]$ and $U:=\{x \in X \mid g(x) \neq 0\}$, then $U$ has the natural structure of an affine variety, and the natural morphism of $U$ into $X$ induces an isomorphism $T_{p} U \rightarrow T_{p} X$ for all $p \in U$.

(iii) For all $s \geqslant 0$, the subset $\left\{x \in X \mid \operatorname{dim} T_{x} X \geqslant s\right\}$ is a Zariski-closed subvariety of $X$.

(b) Show that the set of nilpotent $2 \times 2$ matrices

$X=\left\{x \in \operatorname{Mat}_{2}(k) \mid x^{2}=0\right\}$

may be realised as an affine surface in $\mathbf{A}^{3}$, and determine its tangent space at all points $x \in X$.

Define what it means for two varieties $Y_{1}$ and $Y_{2}$ to be birationally equivalent, and show that the variety $X$ of nilpotent $2 \times 2$ matrices is birationally equivalent to $\mathbf{A}^{2}$.

Paper 2, Section II, I

commentLet $k$ be a field, $J$ an ideal of $k\left[x_{1}, \ldots, x_{n}\right]$, and let $R=k\left[x_{1}, \ldots, x_{n}\right] / J$. Define the radical $\sqrt{J}$ of $J$ and show that it is also an ideal.

The Nullstellensatz says that if $J$ is a maximal ideal, then the inclusion $k \subseteq R$ is an algebraic extension of fields. Suppose from now on that $k$ is algebraically closed. Assuming the above statement of the Nullstellensatz, prove the following.

(i) If $J$ is a maximal ideal, then $J=\left(x_{1}-a_{1}, \ldots, x_{n}-a_{n}\right)$, for some $\left(a_{1}, \ldots, a_{n}\right) \in k^{n}$.

(ii) If $J \neq k\left[x_{1}, \ldots, x_{n}\right]$, then $Z(J) \neq \emptyset$, where

$Z(J)=\left\{a \in k^{n} \mid f(a)=0 \text { for all } f \in J\right\}$

(iii) For $V$ an affine subvariety of $k^{n}$, we set

$I(V)=\left\{f \in k\left[x_{1}, \ldots, x_{n}\right] \mid f(a)=0 \text { for all } a \in V\right\}$

Prove that $J=I(V)$ for some affine subvariety $V \subseteq k^{n}$, if and only if $J=\sqrt{J}$.

[Hint. Given $f \in J$, you may wish to consider the ideal in $k\left[x_{1}, \ldots, x_{n}, y\right]$ generated $b y J$ and $y f-1$.]

(iv) If $A$ is a finitely generated algebra over $k$, and $A$ does not contain nilpotent elements, then there is an affine variety $V \subseteq k^{n}$, for some $n$, with $A=k\left[x_{1}, \ldots, x_{n}\right] / I(V)$.

Assuming $\operatorname{char}(k) \neq 2$, find $\sqrt{J}$ when $J$ is the ideal $\left(x(x-y)^{2}, y(x+y)^{2}\right)$ in $k[x, y]$.

Paper 3, Section II, I

commentLet $X \subset \mathbf{P}^{2}(\mathbf{C})$ be the projective closure of the affine curve $y^{3}=x^{4}+1$. Let $\omega$ denote the differential $d x / y^{2}$. Show that $X$ is smooth, and compute $v_{p}(\omega)$ for all $p \in X$.

Calculate the genus of $X$.

Paper 4, Section II, I

commentLet $X$ be a smooth projective curve of genus 2, defined over the complex numbers. Show that there is a morphism $f: X \rightarrow \mathbf{P}^{1}$ which is a double cover, ramified at six points.

Explain briefly why $X$ cannot be embedded into $\mathbf{P}^{2}$.

For any positive integer $n$, show that there is a smooth affine plane curve which is a double cover of $\mathbf{A}^{1}$ ramified at $n$ points.

[State clearly any theorems that you use.]

Paper 1, Section II, H

comment(i) Let $X$ be an affine variety over an algebraically closed field. Define what it means for $X$ to be irreducible, and show that if $U$ is a non-empty open subset of an irreducible $X$, then $U$ is dense in $X$.

(ii) Show that $n \times n$ matrices with distinct eigenvalues form an affine variety, and are a Zariski open subvariety of affine space $\mathbb{A}^{n^{2}}$ over an algebraically closed field.

(iii) Let $\operatorname{char}_{A}(x)=\operatorname{det}(x I-A)$ be the characteristic polynomial of $A$. Show that the $n \times n$ matrices $A$ such that $\operatorname{char}_{A}(A)=0$ form a Zariski closed subvariety of $\mathbb{A}^{n^{2}}$. Hence conclude that this subvariety is all of $\mathbb{A}^{n^{2}}$.

Paper 2, Section II, H

comment(i) Let $k$ be an algebraically closed field, and let $I$ be an ideal in $k\left[x_{0}, \ldots, x_{n}\right]$. Define what it means for $I$ to be homogeneous.

Now let $Z \subseteq \mathbb{A}^{n+1}$ be a Zariski closed subvariety invariant under $k^{*}=k-\{0\}$; that is, if $z \in Z$ and $\lambda \in k^{*}$, then $\lambda z \in Z$. Show that $I(Z)$ is a homogeneous ideal.

(ii) Let $f \in k\left[x_{1}, \ldots, x_{n-1}\right]$, and let $\Gamma=\left\{(x, f(x)) \mid x \in \mathbb{A}^{n-1}\right\} \subseteq \mathbb{A}^{n}$ be the graph of $f$.

Let $\bar{\Gamma}$ be the closure of $\Gamma$ in $\mathbb{P}^{n}$.

Write, in terms of $f$, the homogeneous equations defining $\bar{\Gamma}$.

Assume that $k$ is an algebraically closed field of characteristic zero. Now suppose $n=3$ and $f(x, y)=y^{3}-x^{2} \in k[x, y]$. Find the singular points of the projective surface $\bar{\Gamma}$.

Paper 3, Section II, H

commentLet $X$ be a smooth projective curve over an algebraically closed field $k$ of characteristic 0 .

(i) Let $D$ be a divisor on $X$.

Define $\mathcal{L}(D)$, and show $\operatorname{dim} \mathcal{L}(D) \leqslant \operatorname{deg} D+1$.

(ii) Define the space of rational differentials $\Omega_{k(X) / k}^{1}$.

If $p$ is a point on $X$, and $t$ a local parameter at $p$, show that $\Omega_{k(X) / k}^{1}=k(X) d t$.

Use that equality to give a definition of $v_{p}(\omega) \in \mathbb{Z}$, for $\omega \in \Omega_{k(X) / k}^{1}, p \in X$. [You need not show that your definition is independent of the choice of local parameter.]

Paper 4, Section II, H

commentLet $X$ be a smooth projective curve over an algebraically closed field $k$.

State the Riemann-Roch theorem, briefly defining all the terms that appear.

Now suppose $X$ has genus 1 , and let $P_{\infty} \in X$.

Compute $\mathcal{L}\left(n P_{\infty}\right)$ for $n \leqslant 6$. Show that $\phi_{3 P_{\infty}}$ defines an isomorphism of $X$ with a smooth plane curve in $\mathbb{P}^{2}$ which is defined by a polynomial of degree 3 .

Paper 1, Section II, G

comment(i) Let $X=\left\{(x, y) \in \mathbb{C}^{2} \mid x^{2}=y^{3}\right\}$. Show that $X$ is birational to $\mathbf{A}^{1}$, but not isomorphic to it.

(ii) Let $X$ be an affine variety. Define the dimension of $X$ in terms of the tangent spaces of $X$.

(iii) Let $f \in k\left[x_{1}, \ldots, x_{n}\right]$ be an irreducible polynomial, where $k$ is an algebraically closed field of arbitrary characteristic. Show that $\operatorname{dim} Z(f)=n-1$.

[You may assume the Nullstellensatz.]

Paper 2, Section II, G

commentLet $X=X_{n, m, r}$ be the set of $n \times m$ matrices of rank at most $r$ over a field $k$. Show that $X_{n, m, r}$ is naturally an affine subvariety of $\mathbf{A}^{n m}$ and that $X_{n, m, r}$ is a Zariski closed subvariety of $X_{n, m, r+1}$.

Show that if $r<\min (n, m)$, then 0 is a singular point of $X$.

Determine the dimension of $X_{5,2,1}$.

Paper 3, Section II, G

comment(i) Let $X$ be a curve, and $p \in X$ be a smooth point on $X$. Define what a local parameter at $p$ is.

Now let $f: X \rightarrow Y$ be a rational map to a quasi-projective variety $Y$. Show that if $Y$ is projective, $f$ extends to a morphism defined at $p$.

Give an example where this fails if $Y$ is not projective, and an example of a morphism $f: \mathbb{C}^{2} \backslash\{0\} \rightarrow \mathbf{P}^{1}$ which does not extend to $0 .$

(ii) Let $V=Z\left(X_{0}^{8}+X_{1}^{8}+X_{2}^{8}\right)$ and $W=Z\left(X_{0}^{4}+X_{1}^{4}+X_{2}^{4}\right)$ be curves in $\mathbf{P}^{2}$ over a field of characteristic not equal to 2 . Let $\phi: V \rightarrow W$ be the map $\left[X_{0}: X_{1}: X_{2}\right] \mapsto\left[X_{0}^{2}: X_{1}^{2}: X_{2}^{2}\right]$. Determine the degree of $\phi$, and the ramification $e_{p}$ for all $p \in V$.

Paper 4, Section II, G

commentLet $E \subseteq \mathbf{P}^{2}$ be the projective curve obtained from the affine curve $y^{2}=\left(x-\lambda_{1}\right)\left(x-\lambda_{2}\right)\left(x-\lambda_{3}\right)$, where the $\lambda_{i}$ are distinct and $\lambda_{1} \lambda_{2} \lambda_{3} \neq 0$.

(i) Show there is a unique point at infinity, $P_{\infty}$.

(ii) Compute $\operatorname{div}(x), \operatorname{div}(y)$.

(iii) Show $\mathcal{L}\left(P_{\infty}\right)=k$.

(iv) Compute $l\left(n P_{\infty}\right)$ for all $n$.

[You may not use the Riemann-Roch theorem.]

Paper 1, Section II, G

commentDefine what is meant by a rational map from a projective variety $V \subset \mathbb{P}^{n}$ to $\mathbb{P}^{m}$. What is a regular point of a rational map?

Consider the rational map $\phi: \mathbb{P}^{2}-\rightarrow \mathbb{P}^{2}$ given by

$\left(X_{0}: X_{1}: X_{2}\right) \mapsto\left(X_{1} X_{2}: X_{0} X_{2}: X_{0} X_{1}\right)$

Show that $\phi$ is not regular at the points $(1: 0: 0),(0: 1: 0),(0: 0: 1)$ and that it is regular elsewhere, and that it is a birational map from $\mathbb{P}^{2}$ to itself.

Let $V \subset \mathbb{P}^{2}$ be the plane curve given by the vanishing of the polynomial $X_{0}^{2} X_{1}^{3}+X_{1}^{2} X_{2}^{3}+X_{2}^{2} X_{0}^{3}$ over a field of characteristic zero. Show that $V$ is irreducible, and that $\phi$ determines a birational equivalence between $V$ and a nonsingular plane quartic.

Paper 2, Section II, G

commentLet $V$ be an irreducible variety over an algebraically closed field $k$. Define the tangent space of $V$ at a point $P$. Show that for any integer $r \geqslant 0$, the set $\left\{P \in V \mid \operatorname{dim} T_{V, P} \geqslant r\right\}$ is a closed subvariety of $V$.

Assume that $k$ has characteristic different from 2. Let $V=V(I) \subset \mathbb{P}^{4}$ be the variety given by the ideal $I=(F, G) \subset k\left[X_{0}, \ldots, X_{4}\right]$, where

$F=X_{1} X_{2}+X_{3} X_{4}, \quad G=X_{0} X_{1}+X_{3}^{2}+X_{4}^{2}$

Determine the singular subvariety of $V$, and compute $\operatorname{dim} T_{V, P}$ at each singular point $P$. [You may assume that $V$ is irreducible.]

Paper 3, Section II, G

commentLet $V$ be a smooth projective curve, and let $D$ be an effective divisor on $V$. Explain how $D$ defines a morphism $\phi_{D}$ from $V$ to some projective space. State the necessary and sufficient conditions for $\phi_{D}$ to be finite. State the necessary and sufficient conditions for $\phi_{D}$ to be an isomorphism onto its image.

Let $V$ have genus 2 , and let $K$ be an effective canonical divisor. Show that the morphism $\phi_{K}$ is a morphism of degree 2 from $V$ to $\mathbb{P}^{1}$.

By considering the divisor $K+P_{1}+P_{2}$ for points $P_{i}$ with $P_{1}+P_{2} \nsim K$, show that there exists a birational morphism from $V$ to a singular plane quartic.

[You may assume the Riemann-Roch Theorem.]

Paper 4, Section II, G

commentState the Riemann-Roch theorem for a smooth projective curve $V$, and use it to outline a proof of the Riemann-Hurwitz formula for a non-constant morphism between projective nonsingular curves in characteristic zero.

Let $V \subset \mathbb{P}^{2}$ be a smooth projective plane cubic over an algebraically closed field $k$ of characteristic zero, written in normal form $X_{0} X_{2}^{2}=F\left(X_{0}, X_{1}\right)$ for a homogeneous cubic polynomial $F$, and let $P_{0}=(0: 0: 1)$ be the point at infinity. Taking the group law on $V$ for which $P_{0}$ is the identity element, let $P \in V$ be a point of order 3 . Show that there exists a linear form $H \in k\left[X_{0}, X_{1}, X_{2}\right]$ such that $V \cap V(H)=\{P\}$.

Let $H_{1}, H_{2} \in k\left[X_{0}, X_{1}, X_{2}\right]$ be nonzero linear forms. Suppose the lines $\left\{H_{i}=0\right\}$ are distinct, do not meet at a point of $V$, and are nowhere tangent to $V$. Let $W \subset \mathbb{P}^{3}$ be given by the vanishing of the polynomials

$X_{0} X_{2}^{2}-F\left(X_{0}, X_{1}\right), \quad X_{3}^{2}-H_{1}\left(X_{0}, X_{1}, X_{2}\right) H_{2}\left(X_{0}, X_{1}, X_{2}\right)$

Show that $W$ has genus 4 . [You may assume without proof that $W$ is an irreducible smooth curve.]