Motivic measures of moduli spaces of 1-dimensional sheaves on rational surfaces

TL;DR

This paper investigates the geometric properties of moduli spaces of rank 0 semistable sheaves on rational surfaces, establishing their irreducibility, stable rationality, and calculating their Betti numbers through motivic measures.

Contribution

It demonstrates the irreducibility and stable rationality of these moduli spaces and computes their Betti numbers using motivic measures, advancing understanding of their geometric structure.

Findings

Moduli spaces are irreducible under certain conditions.

Moduli spaces are stably rational.

Betti numbers are computed via motivic measures.

Abstract

We study the moduli space of rank 0 semistable sheaves on some rational surfaces. We show the irreducibility and stable rationality of them under some conditions. We also compute several (virtual) Betti numbers of those moduli spaces by computing their motivic measures.