Birational geometry and K-stability of Fano varieties

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This PhD project is part of a wider project concerned with the interplay between birational geometry and K-stability of Fano varieties. Algebraic varieties are geometric shapes that are defined by polynomial equations; for example. an ordinary sphere is an algebraic variety. Such shapes appear in all areas where scientists study phenomena described by polynomial equations from computer-aided graphic design to cryptography and mathematical biology. Birational geometry reduces the study of any geometric shape to understanding some building blocks which are of pure geometric type. These pure types correspond to Fano varieties (positively curved), Calabi-Yau varieties (flat) and varietes of general type (negatively curved). In recent years, it has become clear that a differential geometric property guaranteeing the existence of certain canonical metrics was equivalent, in the case of Fano varieties, to an algebraic property called K-stability. These are subtle properties, about which we still have too little geometric intuition. Further, we now understand that K-stability holds the key to understanding families of Fano varieties - or how these degenerate. This project will focus on K-stability of explicit Fano 3-folds, and will consider how stability properties vary in deformation families of Fano 3-folds.