Monads for framed torsion-free sheaves on multi-blow-ups of the projective plane

TL;DR

This paper constructs monads for framed torsion-free sheaves on blow-ups of the complex projective plane, proving the moduli space is a smooth algebraic variety and establishing it as a fine moduli space.

Contribution

It introduces monads for these sheaves and demonstrates the smoothness and fine structure of their moduli space, advancing understanding of their geometric properties.

Findings

Moduli space is a smooth algebraic variety.

Existence of a universal monad on the moduli space.

The moduli space is fine, allowing universal families.

Abstract

We construct monads for framed torsion-free sheaves on blow-ups of the complex projective plane at finitely many distinct points. Using these monads we prove that the moduli space of such sheaves is a smooth algebraic variety. Moreover we construct monads for families of such sheaves parameterized by a noetherian scheme S of finite type. A universal monad on the moduli space is introduced and used to prove that the moduli space is fine.