Isomorphism of compactifications of moduli of vector bundles: nonreduced moduli

TL;DR

This paper establishes an isomorphism between the moduli space of admissible semistable pairs and the Gieseker--Maruyama moduli space of semistable torsion-free sheaves on surfaces, revealing their structural equivalence.

Contribution

It constructs a morphism between the moduli functors and proves their isomorphism, connecting different compactifications of vector bundle moduli.

Findings

The functors are isomorphic.

Main components of the moduli schemes are isomorphic.

Provides a bridge between different moduli compactifications.

Abstract

A morphism of the moduli functor of admissible semistable pairs to the Gieseker -- Maruyama moduli functor (of semistable coherent torsion-free sheaves) with the same Hilbert polynomial on the surface, is constructed. It is shown that these functors are isomorphic, and main components of moduli scheme for semistable admissible pairs are isomorphic to main components of the Gieseker -- Maruyama moduli scheme