GENERA OF BRILL-NOETHER CURVES AND STAIRCASE PATHS IN YOUNG TABLEAUX

In this paper, we compute the genus of the variety of linear series of rank r and degree d on a general curve of genus g, with ramification at least a and 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.