Birational geometry of algebraic varieties

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The proposed research deals with topics in Higher dimensional Algebraic Geometry. It is mainly focused on natural questions in the birational geometry of projective varieties such as the study of questions related to the minimal model program. The PI will investigate the existence of minimal models for varieties not of general type, abundance and the termination of flips, the existence of flips for 3-folds in positive characteristic, the geometry of Fano varieties (in particular the boundedness of log-Fanos), the singularities of normal varieties and multiplier ideal sheaves, the pluricanonical maps of algebraic varieties and the birational geometry of irregular varieties.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 complex projective 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.