The Stacks project

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.)


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0G12. Beware of the difference between the letter 'O' and the digit '0'.