Genera of Brill-Noether curves and staircase paths in Young tableaux

Abstract

In this paper, we compute the genus of the variety of linear series of rank r r and degree d d on a general curve of genus g g , with ramification at least α \alpha and β \beta at two given points, when that variety is 1-dimensional. Our proof uses degenerations and limit linear series along with an analysis of random staircase paths in Young tableaux, and produces an explicit scheme-theoretic description of the limit linear series of fixed rank and degree on a generic chain of elliptic curves when that scheme is itself a curve.