Invariants of the Brill–Noether curve

Abstract

Abstract For a projective nonsingular curve of genus g, the Brill–Noether locus W d r ( C ) $W^r_d(C)$ parametrizes line bundles of degree d over C with at least r + 1 (linearly independent) 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. We introduce combinatorial methods that, with help from the theory of limit linear series, allow us to find invariants of Brill–Noether loci. In particular, we explore the first two invariants of the Brill–Noether curve, giving a new way of calculating the genus of this curve and computing its gonality when C has genus 5.