Algebraic Curves

# Algebraic Curves

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B2.10

commentFor each of the following curves $C$

(i) $C=\left\{(x, y) \in \mathbb{A}^{2} \mid x^{3}-x=y^{2}\right\} \quad$ (ii) $C=\left\{(x, y) \in \mathbb{A}^{2} \mid x^{2} y+x y^{2}=x^{4}+y^{4}\right\}$

compute the points at infinity of $\bar{C} \subset \mathbb{P}^{2}$ (i.e. describe $\bar{C} \backslash C$ ), and find the singular points of the projective curve $\bar{C}$.

At which points of $\bar{C}$ is the rational map $\bar{C} \rightarrow \mathbb{P}^{1}$, given by $(X: Y: Z) \mapsto(X: Y)$, not defined? Justify your answer.

B3.10

comment(i) Let $f: X \rightarrow Y$ be a morphism of smooth projective curves. Define the divisor $f^{*}(D)$ if $D$ is a divisor on $Y$, and state the "finiteness theorem".

(ii) Suppose $f: X \rightarrow \mathbb{P}^{1}$ is a morphism of degree 2 , that $X$ is smooth projective, and that $X \neq \mathbb{P}^{1}$. Let $P, Q \in X$ be distinct ramification points for $f$. Show that, as elements of $c l(X)$, we have $[P] \neq[Q]$, but $2[P]=2[Q]$.

B4.9

commentLet $F(X, Y, Z)$ be an irreducible homogeneous polynomial of degree $n$, and write $C=\left\{p \in \mathbb{P}^{2} \mid F(p)=0\right\}$ for the curve it defines in $\mathbb{P}^{2}$. Suppose $C$ is smooth. Show that the degree of its canonical class is $n(n-3)$.

Hence, or otherwise, show that a smooth curve of genus 2 does not embed in $\mathbb{P}^{2}$.

B2.10

comment(a) For which polynomials $f(x)$ of degree $d>0$ does the equation $y^{2}=f(x)$ define a smooth affine curve?

(b) Now let $C$ be the completion of the curve defined in (a) to a projective curve. For which polynomials $f(x)$ of degree $d>0$ is $C$ a smooth projective curve?

(c) Suppose that $C$, defined in (b), is a smooth projective curve. Consider a map $p: C \rightarrow \mathbb{P}^{1}$, given by $p(x, y)=x$. Find the degree and the ramification points of $p$.

B3.10

comment(a) Let $X \subseteq \mathbb{A}^{n}$ be an affine algebraic variety. Define the tangent space $T_{p} X$ for $p \in X$. Show that the set

$\left\{p \in X \mid \operatorname{dim} T_{p} X \geqslant d\right\}$

is closed, for every $d \geqslant 0$.

(b) Let $C$ be an irreducible projective curve, $p \in C$, and $f: C \backslash\{p\} \rightarrow \mathbb{P}^{n}$ a rational map. Show, carefully quoting any theorems that you use, that if $C$ is smooth at $p$ then $f$ extends to a regular map at $p$.

B4.9

commentLet $X$ be a smooth curve of genus 0 over an algebraically closed field $k$. Show that $k(X)=k\left(\mathbb{P}^{1}\right) .$

Now let $C$ be a plane projective curve defined by an irreducible homogeneous cubic polynomial.

(a) Show that if $C$ is smooth then $C$ is not isomorphic to $\mathbb{P}^{1}$. Standard results on the canonical class may be assumed without proof, provided these are clearly stated.

(b) Show that if $C$ has a singularity then there exists a non-constant morphism from $\mathbb{P}^{1}$ to $C$.

B2.10

commentFor $N \geq 1$, let $V_{N}$ be the (irreducible) projective plane curve $V_{N}: X^{N}+Y^{N}+Z^{N}=0$ over an algebraically closed field of characteristic zero.

Show that $V_{N}$ is smooth (non-singular). For $m, n \geq 1$, let $\alpha_{m, n}: V_{m n} \rightarrow V_{m}$ be the morphism $\alpha_{m, n}(X: Y: Z)=\left(X^{n}: Y^{n}: Z^{n}\right)$. Determine the degree of $\alpha_{m, n}$, its points of ramification and the corresponding ramification indices.

Applying the Riemann-Hurwitz formula to $\alpha_{1, n}$, determine the genus of $V_{n}$.

B3.10

commentLet $f=f(x, y)$ be an irreducible polynomial of degree $n \geq 2$ (over an algebraically closed field of characteristic zero) and $V_{0}=\{f=0\} \subset \mathbb{A}^{2}$ the corresponding affine plane curve. Assume that $V_{0}$ is smooth (non-singular) and that the projectivization $V \subset \mathbb{P}^{2}$ of $V_{0}$ intersects the line at infinity $\mathbb{P}^{2}-\mathbb{A}^{2}$ in $n$ distinct points. Show that $V$ is smooth and determine the divisor of the rational differential $\omega=\frac{d x}{f_{y}^{\prime}}$ on $V$. Deduce a formula for the genus of $V$.

B4.9

commentWrite an essay on the Riemann-Roch theorem and some of its applications.

B2.10

commentLet $f: \mathbb{P}^{2} \rightarrow \mathbb{P}^{2}$ be the rational map given by $f\left(X_{0}: X_{1}: X_{2}\right)=\left(X_{1} X_{2}: X_{0} X_{2}\right.$ : $\left.X_{0} X_{1}\right)$. Determine whether $f$ is defined at the following points: $(1: 1: 1),(0: 1: 1),(0:$ $0: 1)$.

Let $C \subset \mathbb{P}^{2}$ be the curve defined by $X_{1}^{2} X_{2}-X_{0}^{3}=0$. Define a bijective morphism $\alpha: \mathbb{P}^{1} \rightarrow C$. Prove that $\alpha$ is not an isomorphism.

B3.10

commentLet $C$ be the projective curve (over an algebraically closed field $k$ of characteristic zero) defined by the affine equation

$x^{5}+y^{5}=1$

Determine the points at infinity of $C$ and show that $C$ is smooth.

Determine the divisors of the rational functions $x, y \in k(C)$.

Show that $\omega=d x / y^{4}$ is a regular differential on $C$.

Compute the divisor of $\omega$. What is the genus of $C$ ?

B4.9

commentWrite an essay on curves of genus one (over an algebraically closed field $k$ of characteristic zero). Legendre's normal form should not be discussed.