The Stacks project

Exercise 111.54.1 (Definitions). Provide definitions of the italicized concepts.

1. a separated scheme,

2. a universally closed morphism of schemes,

3. $A$ dominates $B$ for local rings $A, B$ contained in a common field,

4. the dimension of a scheme $X$,

5. the codimension of an irreducible closed subscheme $Y$ of a scheme $X$,

Exercise 111.54.2 (Results). State something formally equivalent to the fact discussed in the course.

1. The valuative criterion of properness for a morphism $X \to Y$ of varieties for example.

2. The relationship between $\dim (X)$ and the function field $k(X)$ of $X$ for a variety $X$ over a field $k$.

3. Fill in the blank: The category of nonsingular projective curves over $k$ and nonconstant morphisms is anti-equivalent to $\ldots \ldots \ldots $.

4. Noether normalization.

5. Jacobian criterion.

Exercise 111.54.3. Let $k$ be a field. Let $F \in k[X_0, X_1, X_2]$ be a homogeneous form of degree $d$. Assume that $C = V_{+}(F) \subset \mathbf{P}^2_ k$ is a smooth curve over $k$. Denote $i : C \to \mathbf{P}^2_ k$ the corresponding closed immersion.

1. Show that there is a short exact sequence

   \[ 0 \to \mathcal{O}_{\mathbf{P}^2_ k}(-d) \to \mathcal{O}_{\mathbf{P}^2_ k} \to i_*\mathcal{O}_ C \to 0 \]

   of coherent sheaves on $\mathbf{P}^2_ k$: tell me what the maps are and briefly why it is exact.

2. Conclude that $H^0(C, \mathcal{O}_ C) = k$.

3. Compute the genus of $C$.

4. Assume now that $P = (0 : 0 : 1)$ is not on $C$. Prove that $\pi : C \to \mathbf{P}^1_ k$ given by $(a_0 : a_1 : a_2) \mapsto (a_0 : a_1)$ has degree $d$.

5. Assume $k$ is algebraically closed, assume all ramification indices (the “$e_ i$”) are $1$ or $2$, and assume the characteristic of $k$ is not equal to $2$. How many ramification points does $\pi : C \to \mathbf{P}^1_ k$ have?

6. In terms of $F$, what do you think is a set of equations of the set of ramification points of $\pi $?

7. Can you guess $K_ C$?

Exercise 111.54.4. Let $k$ be a field. Let $X$ be a “triangle” over $k$, i.e., you get $X$ by glueing three copies of $\mathbf{A}^1_ k$ to each other by identifying $0$ on the first copy to $1$ on the second copy, $0$ on the second copy to $1$ on the third copy, and $0$ on the third copy to $1$ on the first copy. It turns out that $X$ is isomorphic to $\mathop{\mathrm{Spec}}(k[x, y]/(xy(x + y + 1)))$; feel free to use this. Compute the Picard group of $X$.

Exercise 111.54.5. Let $k$ be a field. Let $\pi : X \to Y$ be a finite birational morphism of curves with $X$ a projective nonsingular curve over $k$. It follows from the material in the course that $Y$ is a proper curve and that $\pi $ is the normalization morphism of $Y$. We have also seen in the course that there exists a dense open $V \subset Y$ such that $U = \pi ^{-1}(V)$ is a dense open in $X$ and $\pi : U \to V$ is an isomorphism.

1. Show that there exists an effective Cartier divisor $D \subset X$ such that $D \subset U$ and such that $\mathcal{O}_ X(D)$ is ample on $X$.

2. Let $D$ be as in (1). Show that $E = \pi (D)$ is an effective Cartier divisor on $Y$.

3. Briefly indicate why

   1. the map $\mathcal{O}_ Y \to \pi _*\mathcal{O}_ X$ has a coherent cokernel $Q$ which is supported in $Y \setminus V$, and

   2. for every $n$ there is a corresponding map $\mathcal{O}_ Y(nE) \to \pi _*\mathcal{O}_ X(nD)$ whose cokernel is isomorphic to $Q$.

4. Show that $\dim _ k H^0(X, \mathcal{O}_ X(nD)) - \dim _ k H^0(Y, \mathcal{O}_ Y(nE))$ is bounded (by what?) and conclude that the invertible sheaf $\mathcal{O}_ Y(nE)$ has lots of sections for large $n$ (why?).