Euler characteristics of Brill-Noether varieties

TL;DR

This paper derives a formula for the algebraic Euler characteristic of Brill-Noether varieties, connecting algebraic geometry with combinatorial enumeration of set-valued tableaux, using degeneration techniques.

Contribution

It provides a new enumerative formula for the Euler characteristic of Brill-Noether varieties with ramification, utilizing limit linear series and combinatorial enumeration.

Findings

Euler characteristic expressed as count of set-valued tableaux

Uses degeneration methods to relate geometry and combinatorics

Provides explicit enumeration for specific ramification conditions

Abstract

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.