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111.51 Schemes, Final Exam, Fall 2007

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

Exercise 111.51.1 (Definitions). Provide definitions of the following concepts.

  1. $X$ is a scheme

  2. the morphism of schemes $f : X \to Y$ is finite

  3. the morphisms of schemes $f : X \to Y$ is of finite type

  4. the scheme $X$ is Noetherian

  5. the ${\mathcal O}_ X$-module ${\mathcal L}$ on the scheme $X$ is invertible

  6. the genus of a nonsingular projective curve over an algebraically closed field

Exercise 111.51.2. 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 $D(x)$.

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

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

Exercise 111.51.3. Suppose that $X \to \mathop{\mathrm{Spec}}(R)$ is a proper morphism and that $R$ is a discrete valuation ring with residue field $k$. Suppose that $X \times _{\mathop{\mathrm{Spec}}(R)} \mathop{\mathrm{Spec}}(k)$ is the empty scheme. Show that $X$ is the empty scheme.

Exercise 111.51.4. Consider the projective1 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

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

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

  2. Show that $X$ is a nonsingular projective curve.

  3. 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 $3$.

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

  5. Compute the genus of $X$.

Exercise 111.51.5. Let $X \to \mathop{\mathrm{Spec}}({\mathbf Z})$ be a morphism of finite type. Suppose that there is an infinite number of primes $p$ such that $X \times _{\mathop{\mathrm{Spec}}({\mathbf Z})} \mathop{\mathrm{Spec}}({\mathbf F}_ p)$ is not empty.

  1. Show that $X \times _{\mathop{\mathrm{Spec}}({\mathbf Z})}\mathop{\mathrm{Spec}}(\mathbf{Q})$ is not empty.

  2. Do you think the same is true if we replace the condition “finite type” by the condition “locally of finite type”?

[1] The projective embedding is $((X_0, X_1), (Y_0, Y_1))\mapsto (X_0Y_0, X_0Y_1, X_1Y_0, X_1Y_1)$ in other words $(x, y)\mapsto (1, y, x, xy)$.

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