Birational geometry of higher dimensional varieties

University Of Utah, Salt Lake City UT

Investigators

Abstract

The proposed research deals with topics in Higher Dimensional Complex Algebraic Geometry (i.e. the study of the solutions of sets of polynomial equations in several complex variables). Two varieties (i.e. irreducible sets of solutions) are birational if they have isomorphic open subsets. The main focus of this project is on natural questions in the birational geometry of complex projective varieties and in particular on problems related to the Minimal Model Program. The birational classification of surfaces was understood by the Italian school at the beginning of the twentieth century. If a surface is not covered by rational curves then there is a natural choice of a birational surface (known as the minimal model) which has many useful properties. The Minimal Model Program aims to generalize these results to higher dimensions. This program is complete in dimension 3 (by work of Mori, Kawamata, Kollar, Reid, Shokurov and others) and is known to work for varieties of general type (by work of Birkar, Cascini, Hacon, McKernan, Siu and others). This project hopes to answer some of the remaining questions and conjectures that naturally arise in this context.