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

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

2003-05-20

arxiv.org/abs/math/0305286v2

Mathematics

Algebraic Geometry

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

Scientific paper

10.1007/s00222-003-0350-3

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 $Y \subsetneq X$, 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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