Admissible Pairs vs Gieseker--Maruyama

TL;DR

This paper constructs an isomorphism between the moduli functor of admissible semistable pairs and the Gieseker--Maruyama moduli functor, establishing their equivalence and identical moduli schemes for semistable sheaves on surfaces.

Contribution

It proves the equivalence of two moduli functors and their schemes, connecting admissible pairs with Gieseker--Maruyama moduli of semistable sheaves.

Findings

The functors are isomorphic.

The moduli schemes are isomorphic.

All components of the moduli functors are considered.

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 the moduli scheme for semistable admissible pairs $((\tilde S, \tilde L), \tilde E)$ is isomorphic to the Gieseker -- Maruyama moduli scheme. The considerations involve all components of moduli functors and corresponding moduli scheme as they exist.