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

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We generalize the notions of F-regular and F-pure rings to pairs \((R,\mathfrak{a}^t)\) of rings R and ideals \(\mathfrak{a} \subset{R}\) 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 a sort of Inversion of Adjunction of arbitrary codimension, which states that for a pair (X,Y) of a smooth variety X and a closed subscheme \(Y\subsetneq{X}\), if the restriction (Z,Y| Z ) to a normal ℚ-Gorenstein closed subvariety \(Z\subsetneq{X}\) is klt (resp. lc), then the pair (X,Y+Z) is plt (resp. lc) near Z.

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  1. Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1, Komaba, Meguro, Tokyo, 153-8914, Japan

    Shunsuke Takagi

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  1. Shunsuke Takagi
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Correspondence to Shunsuke Takagi.

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Takagi, S. F-singularities of pairs and Inversion of Adjunction of arbitrary codimension. Invent. math. 157, 123–146 (2004). https://doi.org/10.1007/s00222-003-0350-3

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  • DOI: https://doi.org/10.1007/s00222-003-0350-3

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