Invariants of the Brill–Noether curve

Abstract

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.

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

“…In recent work, Farkas and Tarasca [8] also consider a similar problem where a single ramification point is allowed to move. Our proof generalizes the proof given in [3] in the case of trivial ramification and makes the role of skew tableaux explicit. Theorem 1.3.…”

“…Figure 1. Skew shape σ(g, r, d, α, β) associated to the data (g, r, d) = (6,3,4), α=(0,0,2,3), β=(2,1,0,0).…”

“…The original proofs of Theorem 1.1 were based on computations of cohomology classes in the Jacobian. Our proofs use degeneration and limit linear series building on the techniques introduced by Castorena-López-Teixidor in [3], who considered the case r = 1 and no ramification points. We enumerate components of the space of limit linear series on a chain of elliptic curves according to ramification data at the nodes.…”

“…Moreover, the description of the locus of limit linear series on elliptic chains is very explicit and allows to compute invariants of the Brill-Noether locus on the generic curve that are finer than the genus. Indeed, the aim of [3] was to find the gonality of the G 1 4 (X) where X is a generic curve of genus 5.…”

“…A staircase path in a standard Young tableau of shape(5,4,3). The left and right turns are indicated with solid and open dots respectively.…”