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?