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Log geometry and extremal contractions

The Minimal Model Program (in short, MMP) aims at classifying projective algebraic varieties from a birational point of view. That means that starting from a projective algebraic variety X, it is allowed to change the variety under scrutiny as long as its field of rational functions remains the same. In this thesis we study two problems that are inspired by the techniques developed in the last 30 years by various mathematicians in an attempt to realize the Minimal Model Program for varieties of any dimension. In the first part of the thesis, we prove a result about the existence and distribution of rational curves in projective algebraic varieties. We consider projective log canonical pairs (X, A) where the locus Nklt(X, A) of maximal singularities of the pair (X, A) is nonempty. We show that if Kx + A is not nef then there exists an algebraic curves C, whose normalization is isomorphic to A 1, contained either in X \ Nklt(X, A) or in certain locally closed varieties that stratify Nklt(X, A). This result implies a strengthening of the Cone Theorem for log canonical pairs. In the second part, we study certain varieties that naturally arise as possible outcomes of the classification algorithm proposed by the MMP. These are called Mori fibre spaces. A Mori fibre space is a variety X with log canonical singularities together with a morphism f : X -+ Y such that the general fiber of f is a positive dimensional Fano variety and the monodromy of f is as large as possible. We show that being the general fiber of a Mori fiber space is a very restrictive condition for Fano varieties with terminal Q-factorial singularities. More specifically, we obtain two criteria (one sufficient and one necessary) for a Q-factorial Fano variety with terminal singularities to be realized as a fiber of a Mori fiber space. We apply our criteria to figure out what Fano varieties satisfy this property up to dimension three and to study the case of certain homogeneous spaces. The smooth toric case is studied and an interesting connection with K-semistability is also investigated. Thesis Supervisor: James McKernan, FRS Title: Professor of Mathematics University of California, San Diego