---
abstract: 'We define monads for framed torsion-free sheaves on Hirzebruch surfaces and use them to construct moduli spaces for these objects. These moduli spaces are smooth algebraic varieties, and we show that they are fine by constructing a universal monad.'
title: |
    MONADS FOR FRAMED SHEAVES\
    ON HIRZEBRUCH SURFACES
---

[^1]

[Claudio Bartocci,$^\P$ Ugo Bruzzo,$^{\S\ddag}$ and Claudio L. S. Rava$^\P$ ]{}\
$^\P$Dipartimento di Matematica, Università di Genova,\
Via Dodecaneso 35, 16146 Genova, Italia\
$^{\S}$Scuola Internazionale Superiore di Studi Avanzati ([sissa]{}),\
Via Bonomea 265, 34136 Trieste, Italia\
$^\ddag$Istituto Nazionale di Fisica Nucleare, Sezione di Trieste

[ ]{}

Introduction
============

Moduli spaces of framed sheaves over projective surfaces have been the object of some interest over the last few years. In the case of the complex projective plane — generalizing the classical result in [@Do] — these moduli spaces are resolutions of singularities of the moduli space of ideal instantons on the four-sphere $S^4$ [@Nakabook], and as such, they have been used to compute Nekrasov’s partition function, i.e., the partition function of a (suitably twisted) $N=2$ topological super Yang-Mills theory (see [@Ne; @BFMT] and also [@GaLiu] for the more general case of toric surfaces).

More generally, they are at the basis 