108.57 Schemes, Final Exam, Spring 2014

These were the questions in the final exam of a course on Schemes, in the Spring of 2014 at Columbia University.

Exercise 108.57.1 (Definitions). Let $(X, \mathcal{O}_ X)$ be a scheme. Provide definitions of the italicized concepts.

1. the local ring of $X$ at a point $x$,

2. a quasi-coherent sheaf of $\mathcal{O}_ X$-modules,

3. a coherent sheaf of $\mathcal{O}_ X$-modules (please assume $X$ is locally Noetherian,

4. an affine open of $X$,

5. a finite morphism of schemes $X \to Y$.

Exercise 108.57.2 (Theorems). Precisely state a nontrivial fact discussed in the lectures related to each item.

1. on birational invariance of pluri-genera of varieties,

2. being an affine morphism is a local property,

3. the topology of a scheme theoretic fibre of a morphism, and

4. valuative criterion of properness.

Exercise 108.57.3. Let $X = \mathbf{A}^2_\mathbf {C}$ where $\mathbf{C}$ is the field of complex numbers. A line will mean a closed subscheme of $X$ defined by one linear equation $ax + by + c = 0$ for some $a, b, c \in \mathbf{C}$ with $(a, b) \not= (0, 0)$. A curve will mean an irreducible (so nonempty) closed subscheme $C \subset X$ of dimension $1$. A quadric will mean a curve defined by one quadratic equation $ax^2 + bxy + cy^2 + dx + ey + f = 0$ for some $a, b, c, d, e, f \in \mathbf{C}$ and $(a, b, c) \not= (0, 0, 0)$.

1. Find a curve $C$ such that every line has nonempty intersection with $C$.

2. Find a curve $C$ such that every line and every quadric has nonempty intersection with $C$.

3. Show that for every curve $C$ there exists another curve such that $C \cap C' = \emptyset$.

Exercise 108.57.4. Let $k$ be a field. Let $b : X \to \mathbf{A}^2_ k$ be the blow up of the affine plane in the origin. In other words, if $\mathbf{A}^2_ k = \mathop{\mathrm{Spec}}(k[x, y])$, then $X = \text{Proj}(\bigoplus _{n \geq 0} \mathfrak m^ n)$ where $\mathfrak m = (x, y) \subset k[x, y]$. Prove the following statements

1. the scheme theoretic fibre $E$ of $b$ over the origin is isomorphic to $\mathbf{P}^1_ k$,

2. $E$ is an effective Cartier divisor on $X$,

3. the restriction of $\mathcal{O}_ X(-E)$ to $E$ is a line bundle of degree $1$.

(Recall that $\mathcal{O}_ X(-E)$ is the ideal sheaf of $E$ in $X$.)

Exercise 108.57.5. Let $k$ be a field. Let $X$ be a projective variety over $k$. Show there exists an affine variety $U$ over $k$ and a surjective morphism of varieties $U \to X$.

Exercise 108.57.6. Let $k$ be a field of characteristic $p > 0$ different from $2,3$. Consider the closed subscheme $X$ of $\mathbf{P}^ n_ k$ defined by

$\sum \nolimits _{i = 0, \ldots , n} X_ i = 0,\quad \sum \nolimits _{i = 0, \ldots , n} X_ i^2 = 0,\quad \sum \nolimits _{i = 0, \ldots , n} X_ i^3 = 0$

For which pairs $(n, p)$ is this variety singular?

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).