Partial Local Resolution by Characteristic Zero Methods

Strong resolution of singularities in characteristic zero

Hironaka's spectacular proof of resolution of singularities is built on a multiple and intricate induction argument. It is so involved that only few people could really understand it. The constructive proofs given later by Villamayor, Bierstone-Milman and Encinas-Villamayor presented important steps towards a better understanding of the reasoning. They describe an algorithmic procedure for reso...

Annihilators of Local Cohomology in Characteristic Zero

This paper discusses the problem of whether it is possible to annihilate elements of local cohomology modules by elements of arbitrarily small order under a fixed valuation. We first discuss the general problem and its relationship to the Direct Summand Conjecture, and next present two concrete examples where annihilators with small order are shown to exist. We then prove a more general theorem...

Chapter 1 Characteristic p methods in characteristic zero via ultraproducts

Characteristic p methods in characteristic zero via ultraproducts

In recent decades, by exploiting the algebraic properties of the Frobenius in positive characteristic, many so-called homological conjectures and intersection conjectures have been established, culminating into the powerful theory of tight closure and big Cohen–Macaulay algebras. In the present article, I give a survey of how these methods also can be applied directly in characteristic zero by ...

Why the characteristic zero proof of resolution of singularities fails in positive characteristic

This is – for the time being – the last of a series of papers of the author on resolution of singularities. This series started with a collection of obstacles which make resolution in arbitrary dimension and characteristic difficult [Ha 1]. It was followed by a comprehensive study of Hironaka’s proposal for surface resolution in positive characteristic [Ha 2], in order to see whether this appro...