F-singularities of Pairs and Inversion of Adjunction of Arbitrary Codimension

We generalize the notions of F-regular and F-pure rings to those for a pair (R, a) of a ring R and an ideal a ⊂ R with real exponent t > 0, and investigate those properties. Those “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.