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 $r$ and degree $d$ on a general curve of genus $g$, with ramification at least $\alpha$ 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.

36 pages. v2: Re-organized and improved exposition. To appear in Transactions of the AMS

References in corpus (1)

• The Brill-Noether curve and Prym-Tyurin varieties

Cited by in corpus (8)

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• Euler characteristics of Brill-Noether varieties
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• Prym-Brill-Noether Loci of special curves
• On the $q$-Enumeration of Barely Set-Valued Tableaux and Plane Partitions
• Brill-Noether loci of pencils with prescribed ramification on moduli of curves and on Severi varieties on $K3$ surfaces