Globally F-regular and log Fano varieties

By Karl Schwede, Karen E. Smith. 2009

164 citations recorded by the source corpus.

Abstract We prove that every globally F-regular variety is log Fano. In other words, if a prime characteristic variety X is globally F-regular, then it admits an effective Q -divisor Δ such that − K X − Δ is ample and ( X , Δ ) has controlled (Kawamata log terminal, in fact globally F-regular) singularities. A weak form of this result can be viewed as a prime characteristic analog of de Fernex and Hacon's new point of view on Kawamata log terminal singularities in the non- Q -Gorenstein case. We also prove a converse statement in characteristic zero: every log Fano variety has globally F-regular type. Our techniques apply also to F-split varieties, which we show to satisfy a “log Calabi–Yau” condition. We also prove a Kawamata–Viehweg vanishing theorem for globally F-regular pairs.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/8fb46485f6d3548b24d8daa35c0dd99c38bec838

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Citations in the corpus

On the rationality of Kawamata log terminal singularities in positive characteristic (2017, background): “By standard arguments (replacing B with B + (cid:15)H for 0 1 and a very ample divisor H intersecting S properly), it is enough to show that ( X, S + B ) is F -split over a Zariski-open neighbourhood of f ( S ) (see the proof of [SS10, Theorem 3.9]).”
On images of Mori dream spaces (2011, result): “call that the notions of F-purity and global F-regularity are conjecturally related, via reductions topositive characteristics, to logterminality and varieties of Fanotype in characteristic zero (see [SS10]). Since images of varieties of Fano type in characteristic zero are always of Fano type by [F”
A survey of test ideals (2011, background): “to study the global geometry of schemes was introduced in [MR85] and [RR85]. Also see [Hab80] and [BK05]. F-split projective schemes are closely related to log Calabi-Yau pairs in characteristic zero [SS10]. F-signature: Suppose that (R,m) is a local ring of characteristic p>0 with perfect residu”
On the three dimensional minimal model program in positive characteristic (2013, background): “cally over T. When T = X, this deﬁnition coincides with the original deﬁnition of (X,∆) being strongly F-regular and when T= Spec(k), it coincides with the deﬁnition of a globally F-regular pair (cf. [SS10]). The next result shows that the global F-regularity is a very restrictive condition. Theorem”
Test ideals in non-Q-Gorenstein rings (2009, methodology): “This question was also asked by the author and Karen Smith in [SS09] and was aﬃrmatively answered in the case that X is strongly F -regular and a = R (although the version where a (cid:11) = R can also be easily obtained from the methods of this paper).”