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**Adjoint ideals and a correspondence between log canonicity and \(F\)-purity.**
*(English)*
Zbl 1305.14010

\(F\)-singularities are those defined by the action of Frobenius. One of the main open problems in the study of \(F\)-singularities is a conjectured relation between log canonical singularities (from the minimal model program) and \(F\)-pure singularities (which can be defined in the geometric setting by a local splitting of Frobenius). The author proves several interesting results motivated by this conjecture.

First, in Theorem 2.11, the author shows that the weak ordinarity conjecture implies that log canonical singularities are of dense \(F\)-pure type (see the work of M. Mustaţă and V. Srinivas [Nagoya Math. J. 204, 125–157 (2011; Zbl 1239.14011)]). Other results relating the log canonical and \(F\)-pure threshold are also obtained assuming the weak ordinarity conjecture.

Second, in Corollary 3.4, the author shows that the adjoint ideal restricts to a variant of the test ideal via reduction to characteristic \(p \gg 0\) (answering a question proposed earlier by the article under review’s author in [Math. Z. 259, No. 2, 321–341 (2008; Zbl 1143.13007)].

Finally, in Theorem 4.1, building on the method of D. Hernández in [“\(F\)-purity versus log canonicity for polynomials”, preprint, arXiv:1112.2423], he shows that certain log canonical pairs have dense \(F\)-pure type when the defining equations of the pair are sufficiently general.

First, in Theorem 2.11, the author shows that the weak ordinarity conjecture implies that log canonical singularities are of dense \(F\)-pure type (see the work of M. Mustaţă and V. Srinivas [Nagoya Math. J. 204, 125–157 (2011; Zbl 1239.14011)]). Other results relating the log canonical and \(F\)-pure threshold are also obtained assuming the weak ordinarity conjecture.

Second, in Corollary 3.4, the author shows that the adjoint ideal restricts to a variant of the test ideal via reduction to characteristic \(p \gg 0\) (answering a question proposed earlier by the article under review’s author in [Math. Z. 259, No. 2, 321–341 (2008; Zbl 1143.13007)].

Finally, in Theorem 4.1, building on the method of D. Hernández in [“\(F\)-purity versus log canonicity for polynomials”, preprint, arXiv:1112.2423], he shows that certain log canonical pairs have dense \(F\)-pure type when the defining equations of the pair are sufficiently general.

Reviewer: Karl Schwede (Salt Lake City)