These were the questions in the final exam of a course on Schemes, in the Spring of 2007 at Columbia University.

Exercise 109.51.2. Let $X = \mathop{\mathrm{Spec}}({\mathbf Z}[x, y])$, and let ${\mathcal F}$ be a quasi-coherent ${\mathcal O}_ X$-module. Suppose that ${\mathcal F}$ is zero when restricted to the standard affine open $D(x)$.

Show that every global section $s$ of ${\mathcal F}$ is killed by some power of $x$, i.e., $x^ ns = 0$ for some $n\in {\mathbf N}$.

Do you think the same is true if we do not assume that ${\mathcal F}$ is quasi-coherent?

Exercise 109.51.4. Consider the projective^{1} variety

\[ {\mathbf P}^1 \times {\mathbf P}^1 = {\mathbf P}^1_{{\mathbf C}} \times _{\mathop{\mathrm{Spec}}({\mathbf C})} {\mathbf P}^1_{\mathbf C} \]

over the field of complex numbers ${\mathbf C}$. It is covered by four affine pieces, corresponding to pairs of standard affine pieces of ${\mathbf P}^1_{\mathbf C}$. For example, suppose we use homogeneous coordinates $X_0, X_1$ on the first factor and $Y_0, Y_1$ on the second. Set $x = X_1/X_0$, and $y = Y_1/Y_0$. Then the 4 affine open pieces are the spectra of the rings

\[ {\mathbf C}[x, y], \quad {\mathbf C}[x^{-1}, y], \quad {\mathbf C}[x, y^{-1}], \quad {\mathbf C}[x^{-1}, y^{-1}]. \]

Let $X \subset {\mathbf P}^1 \times {\mathbf P}^1$ be the closed subscheme which is the closure of the closed subset of the first affine piece given by the equation

\[ y^3(x^4 + 1) = x^4 -1. \]

Show that $X$ is contained in the union of the first and the last of the 4 affine open pieces.

Show that $X$ is a nonsingular projective curve.

Consider the morphism $pr_2 : X \to {\mathbf P}^1$ (projection onto the first factor). On the first affine piece it is the map $(x, y) \mapsto x$. Briefly explain why it has degree $3$.

Compute the ramification points and ramification indices for the map $pr_2 : X \to {\mathbf P}^1$.

Compute the genus of $X$.

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