111.46 Curves
Exercise 111.46.1. Let $k$ be an algebraically closed field. Let $X$ be a projective curve over $k$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $s_0, \ldots , s_ n \in H^0(X, \mathcal{L})$ be global sections of $\mathcal{L}$. Prove there is a natural closed subscheme
\[ Z \subset \mathbf{P}^ n \times X \]
such that the closed point $((\lambda _0 : \ldots : \lambda _ n), x)$ is in $Z$ if and only if the section $\lambda _0 s_0 + \ldots + \lambda _ n s_ n$ vanishes at $x$. (Hint: describe $Z$ affine locally.)
Exercise 111.46.2. Let $k$ be an algebraically closed field. Let $X$ be a smooth curve over $k$. Let $r \geq 1$. Show that the closed subset
\[ D \subset X \times X^ r = X^{r + 1} \]
whose closed points are the tuples $(x, x_1, \ldots , x_ r)$ with $x = x_ i$ for some $i$, has an invertible ideal sheaf. In other words, show that $D$ is an effective Cartier divisor. Hints: reduce to $r = 1$ and use that $X$ is a smooth curves to say something about the diagonal (look in books for this).
Exercise 111.46.3. Let $k$ be an algebraically closed field. Let $X$ be a smooth projective curve over $k$. Let $T$ be a scheme of finite type over $k$ and let
\[ D_1 \subset X \times T \quad \text{and}\quad D_2 \subset X \times T \]
be two effective Cartier divisors such that for $t \in T$ the fibres $D_{i, t} \subset X_ t$ are not dense (i.e., do not contain the generic point of the curve $X_ t$). Prove that there is a canonical closed subscheme $Z \subset T$ such that a closed point $t \in T$ is in $Z$ if and only if for the scheme theoretic fibres $D_{1, t}$, $D_{2, t}$ of $D_1$, $D_2$ we have
\[ D_{1, t} \subset D_{2, t} \]
as closed subschemes of $X_ t$. Hints: Show that, possibly after shrinking $T$, you may assume $T = \mathop{\mathrm{Spec}}(A)$ is affine and there is an affine open $U \subset X$ such that $D_ i \subset U \times T$. Then show that $M_1 = \Gamma (D_1, \mathcal{O}_{D_1})$ is a finite locally free $A$-module (here you will need some nontrivial algebra — ask your friends). After shrinking $T$ you may assume $M_1$ is a free $A$-module. Then look at
\[ \Gamma (U \times T, \mathcal{I}_{D_2}) \to M_1 = A^{\oplus N} \]
and you define $Z$ as the closed subscheme cut out by the ideal generated by coefficients of vectors in the image of this map. Explain why this works (this will require perhaps a bit more commutative algebra).
Exercise 111.46.4. Let $k$ be an algebraically closed field. Let $X$ be a smooth projective curve over $k$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $s_0, \ldots , s_ n \in H^0(X, \mathcal{L})$ be global sections of $\mathcal{L}$. Let $r \geq 1$. Prove there is a natural closed subscheme
\[ Z \subset \mathbf{P}^ n \times X \times \ldots \times X = \mathbf{P}^ n \times X^ r \]
such that the closed point $((\lambda _0 : \ldots : \lambda _ n), x_1, \ldots , x_ r)$ is in $Z$ if and only if the section $s_\lambda = \lambda _0 s_0 + \ldots + \lambda _ n s_ n$ vanishes on the divisor $D = x_1 + \ldots + x_ r$, i.e., the section $s_\lambda $ is in $\mathcal{L}(-D)$. Hint: explain how this follows by combining then results of Exercises 111.46.2 and 111.46.3.
Exercise 111.46.5. Let $k$ be an algebraically closed field. Let $X$ be a smooth projective curve over $k$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Show that there is a natural closed subset
\[ Z \subset X^ r \]
such that a closed point $(x_1, \ldots , x_ r)$ of $X^ r$ is in $Z$ if and only if $\mathcal{L}(-x_1 - \ldots -x_ r)$ has a nonzero global section. Hint: use Exercise 111.46.4.
Exercise 111.46.6. Let $k$ be an algebraically closed field. Let $X$ be a smooth projective curve over $k$. Let $r \geq s$ be integers. Show that there is a natural closed subset
\[ Z \subset X^ r \times X^ s \]
such that a closed point $(x_1, \ldots , x_ r, y_1, \ldots , y_ s)$ of $X^ r \times X^ s$ is in $Z$ if and only if $x_1 + \ldots + x_ r - y_1 - \ldots - y_ s$ is linearly equivalent to an effective divisor. Hint: Choose an auxiliary invertible module $\mathcal{L}$ of very high degree so that $\mathcal{L}(-D)$ has a nonvanshing section for any effective divisor $D$ of degree $r$. Then use the result of Exercise 111.46.5 twice.
Exercise 111.46.7. Choose your favorite algebraically closed field $k$. As best as you can determine all possible $\mathfrak g^ r_ d$ that can exist on some curve of genus $7$. While doing this also try to
determine in which cases the $\mathfrak g^ r_ d$ is base point free, and
determine in which cases the $\mathfrak g^ r_ d$ gives a closed embedding in $\mathbf{P}^ r$.
Do the same thing if you assume your curve is “general” (make up your own notion of general – this may be easier than the question above). Do the same thing if you assume your curve is hyperelliptic. Do the same thing if you assume your curve is trigonal (and not hyperelliptic). Etc.
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