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.
$X$ is a scheme
the morphism of schemes $f : X \to Y$ is finite
the morphisms of schemes $f : X \to Y$ is of finite type
the scheme $X$ is Noetherian
the ${\mathcal O}_ X$-module ${\mathcal L}$ on the scheme $X$ is invertible
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 $D(x)$.
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 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 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 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
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 $X$.
\[ {\mathbf P}^1 \times {\mathbf P}^1 = {\mathbf P}^1_{{\mathbf C}} \times _{\mathop{\mathrm{Spec}}({\mathbf C})} {\mathbf P}^1_{\mathbf C} \]
\[ {\mathbf C}[x, y], \quad {\mathbf C}[x^{-1}, y], \quad {\mathbf C}[x, y^{-1}], \quad {\mathbf C}[x^{-1}, y^{-1}]. \]
\[ y^3(x^4 + 1) = x^4 -1. \]
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.
Show that $X \times _{\mathop{\mathrm{Spec}}({\mathbf Z})}\mathop{\mathrm{Spec}}(\mathbf{Q})$ is not empty.
Do you think the same is true if we replace the condition “finite type” by the condition “locally of finite type”?
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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)