Algebraic Varieties, Birational Geometry and the Structure of the Galois Groups

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The project addresses the theory of universal spaces for birational invariants of algebraic varieties. One of the fundamental problems concerns the description of a natural class of quotients of products of projective spaces by the projective actions of abelian groups which held all finite birational invariants of varieties defined over algebraic closures of finite fields. The second more difficult problem which PI is going to address is to find a similar class of universal varieties for finite and then for infinite l-adic invariants of algebraic varieties defined over number fields.The Galois theory of functional fields provides with a series of results allowing to describe the cohomological birational invariants of algebraic varieties through the group cohomology. Thus this project addresses the problems at the interface of algebraic geometry, Galois theory and group theory.The PI is going to relate finite birational invariants of functional fields which are higher dimensional analogues of the nonramified Brauer group to the invariants of special systems of abelian subgroups in finite abelian groups.