Globally F-regular varieties: Applications to vanishing theorems for quotients of Fano varieties
type: Item
instance of: scholarly article
title: Globally F-regular varieties: Applications to vanishing theorems for quotients of Fano varieties
zbMATH Open document ID: 0994.14012
author: K. E. Smith
published in: Michigan Mathematical Journal
publication date: 2002-02-04T00:00:00Z
review text: The main results of this paper are on the vanishing of higher cohomology modules of certain line bundles. The author introduces globally F-regular varieties as follows: a projective variety over a Frobenius-finite (prime characteristic) field is globally F-regular if it admits some section ring that is F-regular. A scheme (in characteristic 0) is of strongly F-regular type if in some standard ``reduction to characteristic \(p\)'' a dense set of closed fibers (of positive prime characteristic) is globally F-regular. The author proves that these varieties satisfy strong vanishing properties. Examples of varieties of globally F-regular type are Fano varieties with rational singularities, projective toric varieties, and geometric invariant quotients. Smith proves equivalent formulations of global F-regularity in terms of stable Frobenius splitting. All the theory and notions are carefully and elegantly explained.
reviewed by: Irena Swanson
Mathematics Subject Classification ID: 14F17
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 14G20
zbMATH DE Number: 1700888
zbMATH Keywords: vanishing theorems
zbMATH Keywords: globally \(F\)-regular varieties
zbMATH Keywords: character \(p\)
zbMATH Keywords: Fano varieties
zbMATH Keywords: stable Frobenius splitting
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1030132733
OpenAlex ID: W1968744064
DOI: 10.1307/MMJ/1030132733
description: scientific article; zbMATH DE number 1700888