109.63 Algebraic Geometry, Final Exam, Spring 2020

These were the questions in the final exam of a course on Algebraic Geometry, in the Spring of 2020 at Columbia University.

Exercise 109.63.1 (Definitions). Provide brief definitions of the italicized concepts.

1. a scheme,

2. a morphism of schemes,

3. a quasi-coherent module on a scheme,

4. a variety over a field $k$,

5. a curve over a field $k$,

6. a finite morphism of schemes,

7. the cohomology of a sheaf of abelian groups $\mathcal{F}$ over a topological space $X$,

8. a dualizing sheaf on a scheme $X$ of dimension $d$ proper over a field $k$, and

9. a rational map from a variety $X$ to a variety $Y$.

Exercise 109.63.2 (Theorems). Precisely but briefly state a nontrivial fact discussed in the lectures related to each item (if there is more than one then just pick one of them).

1. cohomology of abelian sheaves on a Noetherian topological space $X$ of dimension $d$,

2. sheaf of differentials $\Omega ^1_{X/k}$ of a smooth variety over a field $k$,

3. dualizing sheaf $\omega _ X$ of a smooth projective variety $X$ over the field $k$,

4. a smooth proper genus $0$ curve over an algebraically closed field $k$, and

5. the genus of a plane curve of degree $d$.

Exercise 109.63.3. Let $k$ be a field. Let $X$ be a scheme over $k$. Assume $X = X_1 \cup X_2$ is an open covering with $X_1$, $X_2$ both isomorphic to $\mathbf{P}^1_ k$ and $X_1 \cap X_2$ isomorphic to $\mathbf{A}^1_ k$. (Such a scheme exists, for example you can take $\mathbf{P}^1_ k$ with $\infty$ doubled.) Show that $\dim _ k H^1(X, \mathcal{O}_ X)$ is infinite.

Exercise 109.63.4. Let $k$ be an algebraically closed field. Let $Y$ be a smooth projective curve of genus $10$. Find a good lower bound for the genus of a smooth projective curve $X$ such that there exists a nonconstant morphism $f : X \to Y$ which is not an isomorphism.

Exercise 109.63.5. Let $k$ be an algebraically closed field of characteristic $0$. Let

$X : T_0^ d + T_1^ d - T_2^ d = 0 \subset \mathbf{P}^2_ k$

be the Fermat curve of degree $d \geq 3$. Consider the closed points $p = [1 : 0 : 1]$ and $q = [0 : 1 : 1]$ on $X$. Set $D = [p] - [q]$.

1. Show that $D$ is nontrivial in the Weil divisor class group.

2. Show that $d D$ is trivial in the Weil divisor class group. (Hint: try to show that both $d[p]$ and $d[q]$ are the intersection of $X$ with a line in the plane.)

Exercise 109.63.6. Let $k$ be an algebraically closed field. Consider the $2$-uple embedding

$\varphi : \mathbf{P}^2 \longrightarrow \mathbf{P}^5$

In terms of the material/notation in the lectures this is the morphism

$\varphi = \varphi _{\mathcal{O}_{\mathbf{P}^2}(2)} : \mathbf{P}^2 \longrightarrow \mathbf{P}(\Gamma (\mathbf{P}^2, \mathcal{O}_{\mathbf{P}^2}(2)))$

In terms of homogeneous coordinates it is given by

$[a_0 : a_1 : a_2] \longmapsto [a_0^2 : a_0a_1 : a_0a_2 : a_1^2 : a_1a_2 : a_2^2]$

It is a closed immersion (please just use this). Let $I \subset k[T_0, \ldots , T_5]$ be the homogeneous ideal of $\varphi (\mathbf{P}^2)$, i.e., the elements of the homogeneous part $I_ d$ are the homogeneous polynomials $F(T_0, \ldots , T_5)$ of degree $d$ which restrict to zero on the closed subscheme $\varphi (\mathbf{P}^2)$. Compute $\dim _ k I_ d$ as a function of $d$.

Exercise 109.63.7. Let $k$ be an algebraically closed field. Let $X$ be a proper scheme of dimension $d$ over $k$ with dualizing module $\omega _ X$. You are given the following information:

1. $\text{Ext}^ i_ X(\mathcal{F}, \omega _ X) \times H^{d - i}(X, \mathcal{F}) \to H^ d(X, \omega _ X) \xrightarrow {t} k$ is nondegenerate for all $i$ and for all coherent $\mathcal{O}_ X$-modules $\mathcal{F}$, and

2. $\omega _ X$ is finite locally free of some rank $r$.

Show that $r = 1$. (Hint: see what happens if you take $\mathcal{F}$ a suitable module supported at a closed point.)

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