  We study birational morphisms between smooth projective surfaces that respect
a given Poisson structure, with particular attention to induced birational maps
between the (Poisson) moduli spaces of sheaves on those surfaces. In
particular, to any birational morphism, we associate a corresponding "minimal
lift" operation on sheaves of homological dimension <=1, and study its
properties. In particular, we show that minimal lift induces a stratification
of the moduli space of simple sheaves on the codomain by open subspaces of the
moduli space of simple sheaves on the domain, compatibly with the induced
Poisson structures.
