Brill-Noether with Movable Ramification
Rebecca Lehman (Tulane)

The classical Brill-Noether theorems count the dimension of the family of maps from a general curve of genus g to non-degenerate curves of degree d in r-dimensional projective space. They extend immediately to the case when you impose specific ramification conditions at fixed general points.

We wish to impose the weaker but in some ways more natural condition that the map should have the specified ramification at any point at all. We solve the problem completely in dimension 1, prove a closed-form existence criterion and a finiteness result in dimension 2, and give an existence test and weak dimension bounds in the general case.