The Stacks project

109.61 Schemes, Final Exam, Spring 2018

These were the questions in the final exam of a course on schemes, in the Spring of 2018 at Columbia University.

Exercise 109.61.1 (Definitions). Provide brief definitions of the italicized concepts. Let $k$ be an algebraically closed field. Let $X$ be a projective curve over $k$.

  1. a smooth algebra over $k$,

  2. the degree of an invertible $\mathcal{O}_ X$-module on $X$,

  3. the genus of $X$,

  4. the Weil divisor class group of $X$,

  5. $X$ is hyperelliptic, and

  6. the intersection number of two curves on a smooth projective surface over $k$.

Exercise 109.61.2 (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. Riemann-Hurwitz theorem,

  2. Clifford's theorem,

  3. factorization of maps between smooth projective surfaces,

  4. Hodge index theorem, and

  5. Riemann hypothesis for curves over finite fields.

Exercise 109.61.3. Let $k$ be an algebraically closed field. Let $X \subset \mathbf{P}^3_ k$ be a smooth curve of degree $d$ and genus $\geq 2$. Assume $X$ is not contained in a plane and that there is a line $\ell $ in $\mathbf{P}^3_ k$ meeting $X$ in $d - 2$ points. Show that $X$ is hyperelliptic.

Exercise 109.61.4. Let $k$ be an algebraically closed field. Let $X$ be a projective curve with pairwise distinct singular points $p_1, \ldots , p_ n$. Explain why the genus of the normalization of $X$ is at most $-n + \dim _ k H^1(X, \mathcal{O}_ X)$.

Exercise 109.61.5. Let $k$ be a field. Let $X = \mathop{\mathrm{Spec}}(k[x, y])$ be affine $2$ space. Let

\[ I = (x^3, x^2y, xy^2, y^3) \subset k[x, y]. \]

Let $Y \subset X$ be the closed subscheme corresponding to $I$. Let $b : X' \to X$ be the blowing up of the ideal $(x, y)$, i.e., the blow up of affine space at the origin.

  1. Show that the scheme theoretic inverse image $b^{-1}Y \subset X'$ is an effective Cartier divisor.

  2. Given an example of an ideal $J \subset k[x, y]$ with $I \subset J \subset (x, y)$ such that if $Z \subset X$ is the closed subscheme corresponding to $J$, then the scheme theoretic inverse image $b^{-1}Z$ is not an effective Cartier divisor.

Exercise 109.61.6. Let $k$ be an algebraically closed field. Consider the following types of surfaces

  1. $S = C_1 \times C_2$ where $C_1$ and $C_2$ are smooth projective curves,

  2. $S = C_1 \times C_2$ where $C_1$ and $C_2$ are smooth projective curves and the genus of $C_1$ is $> 0$,

  3. $S \subset \mathbf{P}^3_ k$ is a hypersurface of degree $4$, and

  4. $S \subset \mathbf{P}^3_ k$ is a smooth hypersurface of degree $4$.

For each type briefly indicate why or why not the class of surfaces of this type contains rational surfaces.

Exercise 109.61.7. Let $k$ be an algebraically closed field. Let $S \subset \mathbf{P}^3_ k$ be a smooth hypersurface of degree $d$. Assume that $S$ contains a line $\ell $. What is the self square of $\ell $ viewed as a divisor on $S$?


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