Adjoint ideals and a correspondence between log canonicity and \(F\)-purity
type: Item
instance of: scholarly article
title: Adjoint ideals and a correspondence between log canonicity and \(F\)-purity
zbMATH Open document ID: 1305.14010
author: Shunsuke Takagi
published in: Algebra \& Number Theory
publication date: 2013-09-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/1105.0072
reviewed by: Karl Schwede
Mathematics Subject Classification ID: 14F18
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 14B05
zbMATH DE Number: 6207092
zbMATH Keywords: test ideals
zbMATH Keywords: log canonical
zbMATH Keywords: \(F\)-pure
zbMATH Keywords: adjoint ideals
MaRDI profile type: Publication
OpenAlex ID: W2964133471
arXiv ID: 1105.0072
DOI: 10.2140/ANT.2013.7.917
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review text: \(F\)-singularities are those defined by the action of Frobenius. One of the main open problems in the study of \(F\)-singularities is a conjectured relation between log canonical singularities (from the minimal model program) and \(F\)-pure singularities (which can be defined in the geometric setting by a local splitting of Frobenius). The author proves several interesting results motivated by this conjecture.NEWLINENEWLINEFirst, in Theorem 2.11, the author shows that the weak ordinarity conjecture implies that log canonical singularities are of dense \(F\)-pure type (see the work of \textit{M. Mustaţă} and \textit{V. Srinivas} [Nagoya Math. J. 204, 125--157 (2011; Zbl 1239.14011)]). Other results relating the log canonical and \(F\)-pure threshold are also obtained assuming the weak ordinarity conjecture.NEWLINENEWLINESecond, in Corollary 3.4, the author shows that the adjoint ideal restricts to a variant of the test ideal via reduction to characteristic \(p \gg 0\) (answering a question proposed earlier by the article under review's author in [Math. Z. 259, No. 2, 321--341 (2008; Zbl 1143.13007)].NEWLINENEWLINEFinally, in Theorem 4.1, building on the method of \textit{D. Hernández} in [``\(F\)-purity versus log canonicity for polynomials'', preprint, \url{arXiv:1112.2423}], he shows that certain log canonical pairs have dense \(F\)-pure type when the defining equations of the pair are sufficiently general.
description: scientific article; zbMATH DE number 6207092