Singularities in Prime Characteristic

Rings of Singularities

This paper is a slightly revised version of an introduction into singularity theory corresponding to a series of lectures given at the ``Advanced School and Conference on homological and geometrical methods in representation theory'' at the International Centre for Theoretical Physics (ICTP), Miramare - Trieste, Italy, 11-29 January 2010. We show how to associate to a triple of posit...

Globally F -regular and Log Fano Varieties

We prove that every globally F -regular variety is log Fano. In other words, if a prime characteristic variety X is globally F -regular, then it admits an effective Qdivisor ∆ such that −KX −∆ is ample and (X, ∆) has controlled (Kawamata log terminal, in fact globally F -regular) singularities. A weak form of this result can be viewed as a prime characteristic analog of de Fernex and Hacon’s ne...

Vanishing, Singularities and Effective Bounds via Prime Characteristic Local Algebra

On the Commutation of the Test Ideal with Localization and Completion

Let R be a reduced ring that is essentially of finite type over an excellent regular local ring of prime characteristic. Then it is shown that the test ideal of R commutes with localization and, if R is local, with completion, under the additional hypothesis that the tight closure of zero in the injective hull E of the residue field of every local ring of R is equal to the finitistic tight clos...

On generalized reduced representations of restricted Lie superalgebras in prime characteristic

Let $mathbb{F}$ be an algebraically closed field of prime characteristic $p>2$ and $(g, [p])$ a finite-dimensional restricted Lie superalgebra over $mathbb{F}$. It is showed that anyfinite-dimensional indecomposable $g$-module is a module for a finite-dimensional quotient of the universal enveloping superalgebra of $g$. These quotient superalgebras are called the generalized reduced enveloping ...