The Stacks project

Exercise 111.66.1 (Definitions). Provide brief definitions of the italicized concepts.

1. a scheme $X$,

2. a quasi-coherent module on a scheme $X$,

3. the Picard group of a scheme $X$,

4. give a morphism $f : X \to Y$ of schemes and an $\mathcal{O}_ X$-module $\mathcal{F}$, the pushforward $f_*\mathcal{F}$,

5. a closed immersion of schemes, and

6. a variety over a given field $k$.

Exercise 111.66.2 (Theorems). Precisely and succintly state one nontrivial fact discussed in the lectures related to each item (if there is more than one then just pick one of them).

1. the topological space of a scheme $X$,

2. a representability criterion for functors on the category of schemes,

3. the Picard functor for a smooth projective curve $X$ over an algebraically closed field $k$,

4. torsion in the Picard group of a smooth projective curve $X$ over an algebraically closed field $k$,

5. a theorem on the Picard group of a product $X \times Y \times Z$ of smooth projective varieties $X$, $Y$, $Z$ over an algebraically closed field $k$.

Exercise 111.66.3. Let $k$ be a field. Explain why $\mathop{\mathrm{Spec}}(k[x])$ has infinitely many points.

Exercise 111.66.4. Let $k$ be a field and let $x_1, \ldots , x_ n$ be variables. Let $f \in k[x_1, \ldots , x_ n]$ be a nonconstant polynomial. Let $X = \mathop{\mathrm{Spec}}(k[x_1, \ldots , x_ n]/(f))$. What property should $f$ have in order for $X$ to be a variety over $k$?

Exercise 111.66.5. Find a singular point (you do not need to find all of them) on the spectrum of $\mathbf{C}[x, y]/(x^5 - 5xy^4 + 20y - 16)$ viewed as a variety over $\mathbf{C}$.

Exercise 111.66.6. Let $X$ be a smooth projective curve over an algebraically closed field $k$. Let $x \in X(k)$ be a closed point on $X$. Let $U = X \setminus \{ x\} $. What can you say about the kernel of the restriction map $\mathop{\mathrm{Pic}}\nolimits (X) \to \mathop{\mathrm{Pic}}\nolimits (U)$?

Exercise 111.66.7. Let $X$ be a smooth projective curve over an algebraically closed field $k$ of genus $g \geq 100$. What values can $h^0(\mathcal{L}) = \dim _ k H^0(X, \mathcal{L})$ take for $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module of degree $2g - 4$ on $X$? Answer as completely as you can.