Birational Geometry and Topology of singular Fano 3-folds.

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The classification of algebraic varieties up to birational equivalence has long been a fundamental problem of Algebraic Geometry.Two varieties are birationally equivalent if they become isomorphic after removing a small subset. It is possible to produce ever larger varieties by simple birational operations (such as blowing up subvarieties), and hence classifying varieties amounts to finding a best, or ``minimal , representative for a birational equivalence class. The Minimal Model Program (MMP) is a still incomplete project started in the 1970s, which given an algebraic variety X, performs a finite number of elementary steps to produce an end product of pure geometric type. These pure geometric type are minimal models on the one hand, and Fano varieties on the other.Minimal models, as their name indicates, realise the hope of being a best (minimal) match for their equivalence class. Fano varieties are close to projective spaces, and should be thought of as the higher dimensional analogue of the sphere in the Uniformisation theorem for Riemann surfaces. Assuming the MMP, the problem of classification of varieties is reduced to understanding the elementary steps of the MMP and its possible outcomes. There remain a number of open questions to achieve completion of the MMP in higher dimensions. In dimension 3, the MMP was completed in the 80s, yet our understanding of its products is partial at best. Some very natural questions remain unanswered. For instance, since the end product of the MMP is not unique, when are two possible end products of the MMP birational? Is it possible to tell which end products are rational, i.e. birational to projective space? My research aims at answering these questions for Fano 3-folds. The MMP produces varieties that are mildly singular-- in dimension 3, these singularities are isolated points. My research shows that when a Fano has ``many'' singular points it tends to acquire many birational maps to other Fano 3-folds, and therefore behave like projective space. What ``many'' means in this context is topological: a Fano has many singular points if these singular points actually lie on a surface S contained in X that is not a hyperplane section of X. My research project argues that, conversally, if there is no such surface lying on X, X behaves as if it was nonsingular. Surprisingly, for Fanos of small degree, this often implies that they are only birational to very few other Fano 3-folds and are therefore nonrational.