  We prove that moduli spaces of torsion-free sheaves on a projective smooth
complex surface are irreducible, reduced and of the expected dimension,
provided the expected dimension is large enough. Actually we prove more: given
a line bundle on the surface, we show that the number of moduli of sheaves
which have a non-zero twisted (by the chosen line-bundle) endomorphism grows
slower than the expected dimension of the moduli space (for fixed rank and
increasing discriminant). The bounds we get are effective: we concentrate on
the case of rank two and we give a lower bound on the discriminant guaranteeing
that the moduli space is reduced of the expected dimension. We also give an
effective irreducibility result for complete intersections. All of the above
follows from a theorem bounding the dimension of complete subsets of the moduli
space which do not intersect the boundary.
