Abstract. 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 α and β 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.

Abstract. In this paper, we give generating functions for the Betti numbers of M 0,0 (G(k, n), d), the moduli stack of zero pointed genus zero degree d stable maps to the Grassmannian G(k, n) for d = 2 and 3.

For a projective nonsingular curve of genus g, the Brill-Noether locus W r d (C) parametrizes line bundles of degree d over C with at least r + 1 sections. When the curve is generic and the Brill-Noether number ρ(g, r, d) equals 1, one can then talk of the Brill-Noether curve. In this paper, we explore the first two invariants of this curve, giving a new way of calculating the genus of this curve and computing its gonality when C has genus 5.

Let C be a projective and nonsingular curve of genus g. Denote by ω the canonical line bundle on C. Consider the locus B k r,d of stable vector bundles of rank r and degree d with at least k independent sections on C. In this paper we show that when C is generic and under some conditions on the degree and genus, there exists a component B of B k r,d of the expected dimension, such that for a generic vector bundle