Donaldson Type Invariants for Algebraic Surfaces : Transition of Moduli Stacks We are defining and studying an algebro geometric analogue of Donaldson invariants by using moduli spaces of semistable sheaves with arbitrary ranks on a polarized projective surface.We are interested in relations among the invariants which are natural generalizations of the wall crossing formula and the Witten conjecture for classical Donaldson invariants. Our goal is to obtain a weaker version of these relations by systematically using the intrinsic smoothness of moduli spaces. According to the recent excellent work of L. Goettsche H. Nakajima and K. Yoshioka the wall crossing formula for Donaldson invariants of projective surfaces can be deduced from such a weaker result in the rank two case! Preliminaries Parabolic L Bradlow Pairs Geometric Invariant Theory and Enhanced Master Space Obstruction Theories of Moduli Stacks and Master Spaces Virtual Fundamental Classes Invariants