Brill-Noether varieties of k-gonal curves

TL;DR

This paper estimates the dimension of special linear series on k-gonal curves using tropical geometry, proving exactness in certain cases and characterizing when the dimension matches that of a general curve.

Contribution

It introduces a tropical geometry approach to estimate the dimension of $W^r_d(C)$ for k-gonal curves and characterizes cases with equal dimension to general curves.

Findings

Exact dimension estimate for $W^r_d(C)$ when k ≥ g/5 + 2

Upper bound for the dimension in other cases

Complete characterization of cases matching general curve dimension

Abstract

We consider a general curve of fixed gonality k and genus g. We propose an estimate for the dimension of the variety $W^r_d(C)$ of special linear series on C, by solving an analogous problem in tropical geometry. Using work of Coppens and Martens, we prove that this estimate is exactly correct if k is at least g/5 + 2, and is an upper bound in all other cases. We also completely characterize the cases in which $W^r_d(C)$ has the same dimension as for a general curve of genus g.