试卷: 代数几何初步

12024 秋季

1.	Let $k$ be an algebraically closed field. Let $H=\{f(x_0,\dots ,x_n)=0\}\subseteq {\mathbb{P}}_k^n$ be a hypersurface of degree $d$. ∘ Is $H$ a connected scheme? ∘ How many irreducible components can $H$ maximally have? ∘ Compute the dimension of $\Gamma (H,{\mathcal{O}}_H(1))$.
2.	Consider the action of ${\mathbb{C}}^{\ast }$ on ${\mathbb{A}}_{\mathbb{C}}^3\setminus \{0\}$ by $$\lambda \cdot (x_0,x_1,x_2)=(\lambda x_0,\lambda x_1,{\lambda }^2{x}_2).$$Set $X=({\mathbb{A}}_{\mathbb{C}}^3\setminus \{0\})/{\mathbb{C}}^{\ast }$ to be the orbit space. ∘ Show that there is a natural scheme structure on $X$. ∘ Is $X$ nonsingular? If not, find the singular point. ∘ Compute the class group of $X$.
3.	Let $\mathcal{F}$ be a quasi–coherent sheaf on a Noetherian scheme $X$. ∘ Show that the locus $U=\{p\in X\mid {\mathcal{F}}_p\text{ is torsion free}\}$ is an open subset of $X$. ∘ Show that $\mathcal{F}$ is locally free if and only if ${\mathcal{F}}_p$ is a free ${\mathcal{O}}_p$-module for all $p\in X$.
4.	Let $C$ be the intersection of two quadric surfaces given by $$C:\{x_0^2+x_1^2+x_2^2+x_3^2=x_0^2+x_1{x}_2+x_3{x}_1+x_2{x}_3=0\}\subseteq {\mathbb{P}}_{\mathbb{C}}^3.$$ ∘ Compute the Picard group of $C$. ∘ Show there is only trivial morphism from $C$ to the Fermat curve $C_0:\{x_0^3+x_1^3+x_2^3=0\}\subseteq {\mathbb{P}}_{\mathbb{C}}^2$.
5.	Let $k$ be a field. Let $C$ be the image of ${\mathbb{P}}_k^1\to {\mathbb{P}}_k^n$ under the map $$[x_0:x_1]\mapsto [x_0^n:x_0^{n-1}{x}_1:x_0^{n-2}{x}_1^2:\cdots :x_1^n].$$ ∘ Show that $C$ is smooth and is not a complete intersection if $n>2$, i.e. it is not an intersection of $n-1$ hypersurfaces. ∘ Show that $C$ is not contained in any linear subspace of ${\mathbb{P}}_k^n$. ∘ Show that if there exists another smooth curve $C^{\prime }\subseteq {\mathbb{P}}_k^n$ and $C^{\prime }\simeq {\mathbb{P}}_k^1$ is not contained in any linear subspace of ${\mathbb{P}}_k^n$, then there exists a change of coordinates $g\in {\mathrm{GL}}_{n+1}(k)$ such that $g(C)=C^{\prime }$.