For reasons discussed in my post below, I would like to construct a curve (over ) and a **finite** map ) with the following properties:

(1) The map is ramified somewhere. (This will be automatic if is not an isomorphism.) Let be the ramification locus and let be .

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

I think I have such a construction. But my argument that 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 be a generic genus curve in . Let be a generic divisor of degree on , and let . A quick Riemmann-Roch computation shows that has two linearly independent sections, so their ratio gives a map to , which we can normalize so that maps to . So we get a map . The Riemann-Hurwitz formula shows that is ramified at points, call them , , …, . So we want to show that , , …, , , , are linearly independent in .

We have the relation , where is the canonical class and the equation takes place in (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?

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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.