On a new compactification of moduli of vector bundles on a surface, III: Functorial approach

TL;DR

This paper introduces a new compactification of the moduli space of Gieseker-stable vector bundles on a surface, using a functorial approach that extends families to modifications of the surface.

Contribution

It constructs a novel compactification for the moduli of vector bundles on surfaces and defines a functorial framework relating polarized schemes and vector bundles.

Findings

New compactification is birationally related to Gieseker--Maruyama space.

Families are extended via modifications of the surface.

The functorial approach unifies families of polarized schemes and vector bundles.

Abstract

A new compactification for the scheme of moduli for Gieseker-stable vector bundles with prescribed Hilbert polynomial, on the smooth projective polarized surface (S,L), is constructed. Families of locally free sheaves on the surface S are completed by locally free sheaves on schemes which are modifications of S. Gieseker -- Maruyama moduli space has a birational morphism onto the new moduli space. We propose the functor for families of pairs polarized scheme -- vector bundle with moduli space of such type.