of the so-called instanton counting [@NY]. Fine moduli spaces of framed sheaves were constructed by Huybrechts and Lehn [@HL1; @HL2] by introducing a stability condition. Bruzzo and Markushevich [@BruMar] showed that framed sheaves on projective surfaces, even without a stability condition, give rise to fine moduli spaces.

Moduli spaces of framed bundles on the projective plane [were]{} considered by Donaldson [@Do; @Do-Kro], [ who established their isomorphism with the moduli of framed instantons on $S^4$]{}. This was extended by King to the case of the blowup of the projective plane at a point [@Ki] and by Buchdahl [@Bu2] to the case of multiple blowups. The degenerate case (including torsion-free sheaves) was first considered, as already cited, by Nakajima [@Nakabook] for the projective plane, and then by Nakajima and Yoshioka [@NY] for the blown-up plane. The case of multiple blowups was studied by Henni [@Henni].

In this paper we consider torsion-free sheaves on a Hirzebruch surface $\Sigma_n$, for $n> 0$, that are framed to the trivial bundle on a generic line $\ell_\infty$ in $\Sigma_n$. We construct a moduli space for such sheaves by using monads. This allows us to obtain a moduli space which is a smooth quasi-projective variety. These moduli 