# Bleg: Independence of points in Picard group

For reasons discussed in my post below, I would like to construct a curve $\overline{S}$ (over $\mathbb{C}$) and a finite map $\overline{F}: S \to \mathbb{C}$) with the following properties:

(1) The map $\overline{F}$ is ramified somewhere. (This will be automatic if $\overline{F}$ is not an isomorphism.) Let $R \subset \overline{S}$ be the ramification locus and let $S$ be $\overline{S} \setminus R$.

(2) The coordinate ring of $S$ has no nontrivial units. This can be thought of as saying that $R$ is not too large.

I think I have such a construction. But my argument that $S$ has no units is basically that I need a bunch of elements of a Picard group to be linearly independent, and I can’t see any relations between them. Obviously, this needs some help! Details follow:

Let $X$ be a generic genus $2$ curve in $\mathbb{P}^2$. Let $D = (d_1, d_2, d_3)$ be a generic divisor of degree $3$ on $X$, and let $\overline{S} = X \setminus D$. A quick Riemmann-Roch computation shows that $\mathcal{O}(D)$ has two linearly independent sections, so their ratio gives a map to $\mathbb{CP}^1$, which we can normalize so that $D$ maps to $\infty$. So we get a map $\overline{S} \to \mathbb{C}$. The Riemann-Hurwitz formula shows that $\overline{F}$ is ramified at $8$ points, call them $r_1$, $r_2$, …, $r_{8}$. So we want to show that $r_1$, $r_2$, …, $r_{8}$, $d_1$, $d_2$, $d_3$ are linearly independent in $\mathrm{Pic}(X)$.

We have the relation $\sum r_i = 2 \sum d_i + K$, where $K$ is the canonical class and the equation takes place in $\mathrm{Pic}(X)$ (exercise!). I can’t see any reason for any other relations to exist so I suspect that, assuming my choices are made generically, none will. Does anyone know how to prove this?

## 2 thoughts on “Bleg: Independence of points in Picard group”

1. Off the top of my head, if I really wanted to know this I guess I’d try to show that as you move around the space of trigonal genus-2 curves, the monodromy action on the ramification points by the full symmetric group on r_1, … r_8. Then any generic linear relation between the r_i and d_i would have to either involve only the d_i, or only the d_i and sum_j r_j, which can itself be expressed in terms of the d_i as you say.