Betti numbers of Brill-Noether varieties on a general curve

TL;DR

This paper computes the rational cohomology of smooth Brill-Noether varieties and determines the intersection cohomology of their singular counterparts on a general curve, advancing understanding of their topological structure.

Contribution

It provides explicit calculations of cohomology groups for both smooth and singular Brill-Noether loci on a general curve, including intersection cohomology.

Findings

Rational cohomology groups of smooth Brill-Noether varieties are computed.

Intersection cohomology of singular Brill-Noether loci is fully determined.

Results enhance understanding of the topology of Brill-Noether varieties.

Abstract

We compute the rational cohomology groups of the smooth Brill-Noether varieties $G^r_d(C)$, parametrizing linear series of degree $d$ and dimension exactly $r$ on a general curve $C$. As an application, we determine the whole intersection cohomology of the singular Brill-Noether loci $W^r_d(C)$, parametrizing complete linear series on $C$ of degree $d$ and dimension at least $r$.