On some isomorphism of compactifications of moduli scheme of vector bundles

TL;DR

This paper constructs a morphism between two moduli functors related to semistable sheaves and admissible pairs, showing their main components are isomorphic, thus revealing a deep connection between different compactifications of the moduli space.

Contribution

It introduces a morphism linking the Gieseker--Maruyama moduli functor to the moduli functor of admissible pairs, demonstrating their main components are isomorphic.

Findings

Main components of the moduli schemes are isomorphic.

A morphism between the moduli functors is constructed.

The isomorphism reveals a connection between different compactifications.

Abstract

A morphism of the reduced Gieseker -- Maruyama moduli functor (of semistable coherent torsion-free sheaves) to the reduced moduli functor of admissible semistable pairs with the same Hilbert polynomial, is constructed. It is shown that main components of reduced moduli scheme for semistable admissible pairs ((\tilde S, \tilde L), \tilde E) are isomorphic to main components of reduced Gieseker -- Maruyama moduli scheme.