Sheaf stable pairs on projective surfaces and birational geometry

TL;DR

This paper investigates the moduli space of higher rank stable pairs on smooth projective surfaces, linking it to birational geometry and minimal models to analyze its structure and components.

Contribution

It introduces a novel connection between higher rank stable pair moduli spaces and stable minimal models, utilizing birational geometry to study their structure.

Findings

Moduli space is isomorphic to a subscheme of the Quot-scheme.

Established a link between stable pairs and stable minimal models.

Analyzed components of the fiber of the Hilbert-Chow morphism.

Abstract

We study moduli space of higher rank marginally stable pairs (E,s:= (s_1,..., s_r)) consisting of torsion free coherent sheaf E of rank r and r sections (s_1,..., s_r) on a smooth projective surface. Having fixed the Chern character of E, the resulting moduli space is isomorphic to some subscheme of the Quot-scheme parametrising quotient sheaves of appropriate Chern character. We establish a connection between moduli space of higher rank stable pairs and stable minimal models induced by the sheaf E and sections s_i and the relative lc model of base surface, and use birational geometry of minimal models to analyse in detail the components of the fibre of the Hilbert-Chow morphism from the moduli space to the Hilbert scheme of effective Cartier divisors on the base surface.