# Nested Hilbert schemes on surfaces: Virtual fundamental class

## TL;DR

This paper constructs virtual fundamental classes on nested Hilbert schemes of points and curves on surfaces, linking them to Seiberg-Witten, stable, and Donaldson-Thomas theories, and provides explicit formulas for related integrals.

## Contribution

It introduces the construction of virtual fundamental classes on nested Hilbert schemes, connecting them to various enumerative invariants and deriving explicit integral formulas.

## Key findings

- Virtual fundamental classes recover Seiberg-Witten and stable theory classes.
- Integrals over nested Hilbert schemes relate to products of Hilbert schemes.
- Explicit formulas are obtained via Carlsson-Okounkov vertex operator methods.

## Abstract

We construct virtual fundamental classes on nested Hilbert schemes of points and curves in complex nonsingular projective surfaces. These classes recover the virtual classes of Seiberg-Witten theory as well as the (reduced) stable theory, and play a crucial role in the reduced Donaldson-Thomas theory of local-surface-threefolds that we study in [GSY17b] (arXiv:1807.05697). We show that certain integrals against the virtual fundamental classes of punctual nested Hilbert schemes are expressed as integrals over the products of the Hilbert scheme of points. We are able to find explicit formulas for some of these integrals by relating them to Carlsson-Okounkov's vertex operator formulas.