Towards Bounding Complexity of a Minimal Model

We give some effectivity results in birational geometry. We provide an upper bound on the rational constant in Rationality Theorem in terms of certain intersection numbers, under an additional condition on the variety that it admits a divisorial contraction. One consequence is an explicit bound on the number of certain extremal rays. Our main result tries to construct from a given set of ample divisors H_j on X with their intersection numbers b_i, a certain set of ample divisors L_j on X' or X⁺ where X' or X⁺ arises from a contraction or a flip,such that the corresponding intersection numbers of L_j are uniformly bounded in terms of b_i and the index of X. This gives a bound on the projective degree of a minimal model in special case.