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 Hj on X with their intersection numbers bi, a certain set of ample divisors Lj on X' or X+ where X' or X+ arises from a contraction or a flip, such that the corresponding intersection numbers of Lj are uniformly bounded in terms of bi and the index of X. This gives a bound on the projective degree of a minimal model in special case.

In this paper, we consider the preservation of stability by using the notion of twisted stability. As applications, (1) we show that moduli spaces of stable sheaves on K3 and abelian surfaces are irreducible and (2) we compute Hodge polynomials of some moduli spaces of stable sheaves on Enriques surfaces.

This paper is devoted to the calculation of the B-model chiral ring, used in physics, for semiample Calabi–Yau hypersurfaces. We also study the cohomology of semiample hypersurfaces.

Alexander Premet has stated the following problem: what is a relation between subregular nilpotent representations of a classical semisimple restricted Lie algebra and non-commutative deformations of the corresponding singularities? We solve this problem for type A.