The Stacks project

109.54 Schemes, Final Exam, Spring 2011

These were the questions in the final exam of a course on Schemes, in the Spring of 2011 at Columbia University.

Exercise 109.54.1 (Definitions). Provide definitions of the italicized concepts.

  1. a separated scheme,

  2. a universally closed morphism of schemes,

  3. $A$ dominates $B$ for local rings $A, B$ contained in a common field,

  4. the dimension of a scheme $X$,

  5. the codimension of an irreducible closed subscheme $Y$ of a scheme $X$,

Exercise 109.54.2 (Results). State something formally equivalent to the fact discussed in the course.

  1. The valuative criterion of properness for a morphism $X \to Y$ of varieties for example.

  2. The relationship between $\dim (X)$ and the function field $k(X)$ of $X$ for a variety $X$ over a field $k$.

  3. Fill in the blank: The category of nonsingular projective curves over $k$ and nonconstant morphisms is anti-equivalent to $\ldots \ldots \ldots $.

  4. Noether normalization.

  5. Jacobian criterion.

Exercise 109.54.3. Let $k$ be a field. Let $F \in k[X_0, X_1, X_2]$ be a homogeneous form of degree $d$. Assume that $C = V_{+}(F) \subset \mathbf{P}^2_ k$ is a smooth curve over $k$. Denote $i : C \to \mathbf{P}^2_ k$ the corresponding closed immersion.

  1. Show that there is a short exact sequence

    \[ 0 \to \mathcal{O}_{\mathbf{P}^2_ k}(-d) \to \mathcal{O}_{\mathbf{P}^2_ k} \to i_*\mathcal{O}_ C \to 0 \]

    of coherent sheaves on $\mathbf{P}^2_ k$: tell me what the maps are and briefly why it is exact.

  2. Conclude that $H^0(C, \mathcal{O}_ C) = k$.

  3. Compute the genus of $C$.

  4. Assume now that $P = (0 : 0 : 1)$ is not on $C$. Prove that $\pi : C \to \mathbf{P}^1_ k$ given by $(a_0 : a_1 : a_2) \mapsto (a_0 : a_1)$ has degree $d$.

  5. Assume $k$ is algebraically closed, assume all ramification indices (the “$e_ i$”) are $1$ or $2$, and assume the characteristic of $k$ is not equal to $2$. How many ramification points does $\pi : C \to \mathbf{P}^1_ k$ have?

  6. In terms of $F$, what do you think is a set of equations of the set of ramification points of $\pi $?

  7. Can you guess $K_ C$?

Exercise 109.54.4. Let $k$ be a field. Let $X$ be a “triangle” over $k$, i.e., you get $X$ by glueing three copies of $\mathbf{A}^1_ k$ to each other by identifying $0$ on the first copy to $1$ on the second copy, $0$ on the second copy to $1$ on the third copy, and $0$ on the third copy to $1$ on the first copy. It turns out that $X$ is isomorphic to $\mathop{\mathrm{Spec}}(k[x, y]/(xy(x + y + 1)))$; feel free to use this. Compute the Picard group of $X$.

Exercise 109.54.5. Let $k$ be a field. Let $\pi : X \to Y$ be a finite birational morphism of curves with $X$ a projective nonsingular curve over $k$. It follows from the material in the course that $Y$ is a proper curve and that $\pi $ is the normalization morphism of $Y$. We have also seen in the course that there exists a dense open $V \subset Y$ such that $U = \pi ^{-1}(V)$ is a dense open in $X$ and $\pi : U \to V$ is an isomorphism.

  1. Show that there exists an effective Cartier divisor $D \subset X$ such that $D \subset U$ and such that $\mathcal{O}_ X(D)$ is ample on $X$.

  2. Let $D$ be as in (1). Show that $E = \pi (D)$ is an effective Cartier divisor on $Y$.

  3. Briefly indicate why

    1. the map $\mathcal{O}_ Y \to \pi _*\mathcal{O}_ X$ has a coherent cokernel $Q$ which is supported in $Y \setminus V$, and

    2. for every $n$ there is a corresponding map $\mathcal{O}_ Y(nE) \to \pi _*\mathcal{O}_ X(nD)$ whose cokernel is isomorphic to $Q$.

  4. Show that $\dim _ k H^0(X, \mathcal{O}_ X(nD)) - \dim _ k H^0(Y, \mathcal{O}_ Y(nE))$ is bounded (by what?) and conclude that the invertible sheaf $\mathcal{O}_ Y(nE)$ has lots of sections for large $n$ (why?).


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 069Z. Beware of the difference between the letter 'O' and the digit '0'.