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109.59 Schemes, Final Exam, Spring 2017

These were the questions in the final exam of a course on schemes, in the Spring of 2017 at Columbia University.

Exercise 109.59.1 (Definitions). Let $f : X \to Y$ be a morphism of schemes. Provide brief definitions of the italicized concepts.

  1. the scheme theoretic fibre of $f$ at $y \in Y$,

  2. $f$ is a finite morphism,

  3. a quasi-coherent $\mathcal{O}_ X$-module,

  4. $X$ is variety,

  5. $f$ is a smooth morphism,

  6. $f$ is a proper morphism.

Exercise 109.59.2 (Theorems). Precisely but briefly state a nontrivial fact discussed in the lectures related to each item.

  1. pushforward of quasi-coherent sheaves,

  2. cohomology of coherent sheaves on projective varieties,

  3. Serre duality for a projective scheme over a field, and

  4. Riemann-Hurwitz.

Exercise 109.59.3. Let $k$ be an algebraically closed field. Let $\ell > 100$ be a prime number different from the characteristic of $k$. Let $X$ be the nonsingular projective model of the affine curve given by the equation

\[ y^\ell = x(x - 1)^3 \]

in $\mathbf{A}^2_ k$. Answer the following questions:

  1. What is the genus of $X$?

  2. Give an upper bound for the gonality1 of $X$.

Exercise 109.59.4. Let $k$ be an algebraically closed field. Let $X$ be a reduced, projective scheme over $k$ all of whose irreducible components have the same dimension $1$. Let $\omega _{X/k}$ be the relative dualizing module. Show that if $\dim _ k H^1(X, \omega _{X/k}) > 1$, then $X$ is disconnected.

Exercise 109.59.5. Give an example of a scheme $X$ and a nontrivial invertible $\mathcal{O}_ X$-module $\mathcal{L}$ such that both $H^0(X, \mathcal{L})$ and $H^0(X, \mathcal{L}^{\otimes -1})$ are nonzero.

Exercise 109.59.6. Let $k$ be an algebraically closed field. Let $g \geq 3$. Let $X$ and $X'$ be smooth projective curves over $k$ of genus $g$ and $g + 1$. Let $Y \subset X \times X'$ be a curve such that the projections $Y \to X$ and $Y \to X'$ are nonconstant. Prove that the nonsingular projective model of $Y$ has genus $\geq 2g + 1$.

Exercise 109.59.7. Let $k$ be a finite field. Let $g > 1$. Sketch a proof of the following: there are only a finite number of isomorphism classes of smooth projective curves over $k$ of genus $g$. (You will get credit for even just trying to answer this.)

[1] The gonality is the smallest degree of a nonconstant morphism from $X$ to $\mathbf{P}^1_ k$.

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