F-singularities of pairs and Inversion of Adjunction of arbitrary codimension

Abstract

We generalize the notions of F-regular and F-pure rings to pairs $(R,\a^t)$ of rings $R$ and ideals $\a \subset R$ with real exponent $t>0$, and investigate these properties. These ``F-singularities of pairs'' correspond to singularities of pairs of arbitrary codimension in birational geometry. Via this correspondence, we prove Inversion of Adjunction of arbitrary codimension, which states that for a pair $(X,Y)$ of a smooth variety $X$ and a closed subscheme $Y⊊X$, if the restriction $(Z,Y{|}_Z)$ to a normal $\Q$-Gorenstein closed subvariety $Z⊊X$ is klt (resp. lc), then the pair $(X,Y+Z)$ is plt (resp. lc) near $Z$.

21 pages, AMS-LaTeX; v.2: minor changes, to appear in Invent. Math

References in corpus (1)

• Jet schemes, log discrepancies and Inversion of Adjunction

Cited by in corpus (27)

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• Supplements to non-lc ideal sheaves
• Adjoint ideals and a correspondence between log canonicity and F-purity
• Seshadri Constants and Fujita's Conjecture via Positive Characteristic Methods
• On a comparison of minimal log discrepancies in terms of motivic integration
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• Arithmetic and geometric deformations of $F$-pure and $F$-regular singularities
• F-purity of hypersurfaces
• F-thresholds of graded rings
• A Gorenstein criterion for strongly $F$-regular and log terminal singularities
• Inversion of adjunction for $F$-signature
• The Cartier core map for Cartier algebras
• Rational singularities associated to pairs
• $F$-singularities under generic linkage
• Jet schemes of homogeneous hypersurfaces
• Generalized test ideals and symbolic powers
• $F$-pure thresholds and $F$-Volumes of some non principal ideals
• On F-pure thresholds
• F-injective singularities are Du Bois