Towards a non-Q-Gorenstein Minimal Model Program

Abstract

In this thesis we do the first steps towards a non-Q-Gorenstein Minimal Model Program. We extensively study non-Q-factorial singularities, using the techniques introduced by [dFH09]. We introduce a new class of singularities, log terminal+, which we show satisfies several nice properties; we investigate the finite generation of the canonical algebra of local sections, we relate log terminal+ singularities with existing classes, we show a Bertini-type theorem, and small deformation invariance. We also provide a list of examples of the pathologies that can occur when working in the non-Q-factorial setting. We subsequently focus on defining and studying positivity for Weil divisors. We define nefness / amplitude / bigness / pseudo-effectivity for Weil divisors; we show various characterizations of this notions, and we prove vanishing and non-vanishing theorems. We conclude with a proposal of a non-Q-Gorenstein MMP, we prove it for toric varieties, and we discuss where the obstacles lay in the general case. As application of our techniques, we prove the existence on non-Q-factorial log terminal flips.