Maximal subbundles, quot schemes, and curve counting

TL;DR

This paper investigates the relationship between maximal subbundles, Quot schemes, and curve counting in the total space of a rank 2 vector bundle over a genus g curve, connecting stable pairs invariants with enumerative geometry.

Contribution

It establishes a link between stable pairs invariants and curve counting in the total space of vector bundles, extending the virtual intersection theory of Quot schemes.

Findings

Expresses loci of stable pairs as products of Quot schemes.

Relates stable pairs invariants to curve counts in the total space of E.

Provides enumerative interpretation of residue invariants as curve counts.

Abstract

Let $E$ be a rank 2, degree $d$ vector bundle over a genus $g$ curve $C$. The loci of stable pairs on $E$ in class $2[C]$ fixed by the scaling action are expressed as products of $\Quot$ schemes. Using virtual localization, the stable pairs invariants of $E$ are related to the virtual intersection theory of $\Quot E$. The latter theory is extensively discussed for an $E$ of arbitrary rank; the tautological ring of $\Quot E$ is defined and is computed on the locus parameterizing rank one subsheaves. In case $E$ has rank 2, $d$ and $g$ have opposite parity, and $E$ is sufficiently generic, it is known that $E$ has exactly $2^g$ line subbundles of maximal degree. Doubling the zero section along such a subbundle gives a curve in the total space of $E$ in class $2[C]$. We relate this count of maximal subbundles with stable pairs/Donaldson-Thomas theory on the total space of $E$. This endows the residue invariants of $E$ with enumerative significance: they actually \emph{count} curves in $E$.