We prove an enumerative formula for the algebraic Euler characteristic of Brill-Noether varieties, parametrizing degree d and rank r linear series on a general genus g curve, with ramification profiles specified at up to two general points. Up to sign, this Euler characteristic is the number of standard set-valued tableaux of a certain skew shape with g labels. We use a flat degeneration via the Eisenbud-Harris theory of limit linear series, relying on moduli-theoretic advances of Osserman and Murray-Osserman; the count of set-valued tableaux is an explicit enumeration of strata of this degeneration.
- Pub Date:
- August 2017
- Mathematics - Algebraic Geometry;
- Mathematics - Combinatorics;
- 18 pages, 2 figures. v3: minor corrections and updated references. Final version published in Transactions of the AMS