  We apply virtual localization to the problem of finding blowup formulae for
virtual sheaf-theoretic invariants on a smooth projective surface. This leads
to a general procedure that can be used to express virtual enumerative
invariants on the blowup in terms of those on the original surface. We use an
enhanced master space construction over the moduli spaces of $m$-stable sheaves
introduced by Nakajima and Yoshioka. Our work extends their analogous results
for the equivariant moduli spaces of framed sheaves on $\mathbb{P}^2$. In
contrast to their work, we make no use of GIT methods and work with an
arbitrary smooth complex projective surface, assuming only the absence of
strictly semistable sheaves. The main examples to keep in mind are Mochizuki's
virtual analogue of the Donaldson invariant and the virtual $\chi_y$-genus of
the moduli space of Gieseker semistable sheaves on the surface.
