Moduli of sheaves, Fourier-Mukai transform, and partial desingularization

TL;DR

This paper explores birational relationships among various moduli spaces of sheaves and stable maps, utilizing Fourier-Mukai transforms and partial desingularization to understand their geometric properties.

Contribution

It establishes explicit birational maps between moduli spaces of sheaves and stable maps, employing Fourier-Mukai transforms and Kirwan's desingularization techniques.

Findings

Describes a birational morphism via Fourier-Mukai transform.

Identifies partial desingularization of Kontsevich's moduli space.

Analyzes geometric properties through variation of stable pairs.

Abstract

We study birational maps among 1) the moduli space of semistable torsion sheaves of Hilbert polynomial $4m+2$ on a smooth quadric surface, 2) the moduli space of semistable torsion sheaves of Hilbert polynomial $m^{2}+3m+2$ on $\mathbb{P}^{3}$, 3) Kontsevich's moduli space of genus zero stable maps of degree 2 to Grassmannian $Gr(2, 4)$. A regular birational morphism from 1) to 2) is described in terms of Fourier-Mukai transform. The map from 3) to 2) is Kirwan's partial desingularization. Also we investigate several geometric properties of 1) by using the variation of moduli spaces of stable pairs.