# Degeneracy loci, virtual cycles and nested Hilbert schemes II

## TL;DR

This paper develops a method to compute invariants of nested Hilbert schemes and related sheaf-counting theories by expressing them as degeneracy loci and applying virtual cycle techniques, enabling explicit calculations.

## Contribution

It introduces a novel approach to represent nested Hilbert schemes as virtual resolutions of degeneracy loci, facilitating the calculation of various enumerative invariants.

## Key findings

- Provides explicit formulas for invariants like VW, SW, local PT, and local DT.
- Shows how to modify obstruction theories to produce virtual cycles for sheaf-counting.
- Offers an effective computational framework using Thom-Porteous-like Chern class formulas.

## Abstract

We express nested Hilbert schemes of points and curves on a smooth projective surface as "virtual resolutions" of degeneracy loci of maps of vector bundles on smooth ambient spaces. We show how to modify the resulting obstruction theories to produce the virtual cycles of Vafa-Witten theory and other sheaf-counting problems. The result is an effective way of calculating invariants (VW, SW, local PT and local DT) via Thom-Porteous-like Chern class formulae.