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F-singularities of pairs and inversion of adjunction of arbitrary codimension. (English) Zbl 1121.13008

This nicely written article is studying the complexity of singularities of varieties with positive characteristic methods. Firstly, the author generalizes the terminology of F-singularities [N. Hara and K.-i. Watanabe, J. Algebr. Geom. 11, 363–392 (2002; Zbl 1013.13004)] to pairs \((R,\mathbf{a}_1^{t_1}\cdots\mathbf{a}_k^{t_k})\), where \(R\) is a F-finite reduced commutative ring of characteristic \(p>0\), \(\mathbf{a}_i\) are ideals in \(R\), and \(t_i\) are positive real numbers. He also proves an “F-inversion of adjunction”: if \(I\subsetneq R\) is an unmixed radical ideal and \(S=R/I\), then \((S,(\mathbf{a}_1S)^{t_1}\cdots (\mathbf{a}_kS)^{t_k})\) being F-pure (respectively, strongly F-regular) implies that \((R,I\mathbf{a}_1^{t_1}\cdots\mathbf{a}_k^{t_k})\) is F-pure (respectively, purely F-regular). Secondly, he relates the terminology of F-singularities of pairs to the singularities of pairs in characteristic zero [J. Kollár, in: Algebraic geometry. Proc. Symp. Pure Math. 62, 221–287 (1997; Zbl 0905.14002)]) via reductions modulo \(p\): strongly F-regular type corresponds to Kawamata log terminal (klt), purely F-regular type implies purely log terminal (plt), and dense F-pure type implies log canonical (lc). He obtains then a sort of inversion of adjunction in characteristic zero which states that for a pair \((X,Y)\) of a smooth variety \(X\) and a formal linear combination of closed subschemes \(Y=\sum_{t=1}^k t_iY_i\) (\(t_i\in \mathbb R_{>0}\)), if \((Z,Y_{| Z})\) is klt (respectively, lc) then \((X,Y+Z)\) is plt (respectively, lc) near \(Z\), where \(Z\) is a normal \(\mathbb Q\)-Gorenstein closed proper subvariety of \(X\) not included in \(\bigcup_iY_i\). The case when \(Z\) has codimension one was proved by L. Ein, M. Mustaţă, and T. Yasuda [Invent. Math. 153, 519–535 (2003; Zbl 1049.14008)] and this article gives a new proof for this case based on positive characteristic methods.

MSC:

13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
14B05 Singularities in algebraic geometry
14N30 Adjunction problems
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