Hyperquot schemes on curves: virtual class and motivic invariants

TL;DR

This paper constructs a virtual class on hyperquot schemes over curves, enabling the computation of motivic invariants and providing criteria for smoothness and unobstructedness, advancing enumerative geometry tools.

Contribution

It introduces a perfect obstruction theory for hyperquot schemes on curves, allowing for motivic invariant calculations and criteria for smoothness.

Findings

Established a virtual class on hyperquot schemes

Derived motivic partition functions in terms of the motivic zeta function

Provided criteria for smoothness and unobstructedness

Abstract

Let $C$ be a smooth projective curve, $E$ a locally free sheaf. Hyperquot schemes on $C$ parametrise flags of coherent quotients of $E$ with fixed Hilbert polynomial, and offer alternative compactifications to the spaces of maps from $C$ to partial flag varieties. Motivated by enumerative geometry, in this paper we construct a perfect obstruction theory (and hence a virtual class and a virtual structure sheaf) on these moduli spaces, which we use to provide criteria for smoothness and unobstructedness. Under these assumptions, we determine their motivic partition function in the Grothendieck ring of varieties, in terms of the motivic zeta function of $C$.