F-regular and F-pure rings vs. log terminal and log canonical singularities

By N. Hara, Kei-ichi Watanabe. 2000

225 citations recorded by the source corpus.

In this paper, we investigate the relationship of F-regular (resp. F-pure) rings and log terminal (resp. log canonical) singularities. Also, we extend the notions of F-regularity and F-purity to "F-singularities of pairs." The notions of F-regular and F-pure rings in characteristic $p>0$ are characterized by a splitting of the Frobenius map, and define some classes of rings having "mild" singularities. On the other hand, there are notions of log terminal and log canonical singularities defined via resolutions of singularities in characteristic zero. These are defined also for pairs of a normal variety and a $\Bbb Q$-divisor $\Delta$ on it, and play an important role in birational algebraic geometry. As an analog of these singularities of pairs in characteristic zero, we define the notions of "F-singularities of pairs," namely strong F-regularity, divisorial F-regularity and F-purity for a pair $(A,\Delta)$ of a normal ring $A$ of characteristic $p > 0$ and an effective $\Bbb Q$-divisor $\Delta$ on $Y = Spec A$. The main theorem of this paper asserts that, if $K_Y + \Delta$ is $\Bbb Q$-Cartier, then the above three variants of F-singularitiesof pairs imply KLT, PLT and LC properties, respectively. We also prove some results for F-singularities of pairs which are analoguous to singularities of pairs in characteristic zero.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/27adf25eabd6b14b5cb6ab7492a05cac71bb8948

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Citations in the corpus

F-signature and Hilbert–Kunz multiplicity : a combined approach and comparison (2016, unknown): “Together with the characterization of strong F -regularity in terms of the positivity of the F -signature [Aberbach and Leuschke 2003], these two results are at the core of the use of asymptotic Frobenius techniques to measure singularities.”
Hodge theory meets the minimal model program: a survey of log canonical and Du Bois singularities (2009, background): “o be investigated, cf. [Fed83]. After the introduction of tight closure in [HH90], a precise correspondence between several classes of singularities was established. See, for example, [FW89], [MS91], [HW02], [Smi97], [Har98], [MS97], [Smi00], [Har05], [HY03], [Tak04], [TW04], [Tak08]. The second nam”
Integral closure of ideals, rings, and modules (2006, unknown): “and Hara and Watanabe [108].”
A canonical linear system associated to adjoint divisors in characteristic $p > 0$ (2011, background): “ = OX. (iv) we say that Xis F-injective if Φi : hi(Fe ∗ ω q X) −→ hi(ω q X) is surjective for each i. Remark 2.11. Strongly F-regular pairs are the analog of log terminal pairs in characteristic zero [HW02]. F-rational varieties are the analog of varieties with rational singularities in characterist”
A Guided Tour to Normalized Volume (2018, background): “ties are closely related to strongly F-regular singularities in positive characteristic. Moreover, log canonical thresholds (lct) correspond to F-pure thresholds (fpt) in positive characteristic (see [HW02]). In spirit of Theorem 2.6, we deﬁne the F-volume of singularities in characteristic p as fol”
An interpretation of multiplier ideals via tight closure (2001, background): “cently it became clear that F-singularities (F-regular and F-pure rings) correspond to singularities arising in birational geometry (Kawamata log terminal and log canonical singularities). See [Ha2], [HW], [MS], [Sm1]. The notions of Kawamata log terminal (or klt for short) and log canonical (or for”
F-signature of pairs and the asymptotic behavior of Frobenius splittings (2011, methodology): “to vastly generalize the theory of F-signature to so-called pairs. For example, we shall give deﬁnitions for the F-signature of the ideal pairs (R,at) used in [HY03] as well as the divisor pairs from [HW02] that appear throughout birational algebraic geometry. While interesting in its own right, thi”
ON THE BEHAVIOR OF TEST IDEALS UNDER FINITE MORPHISMS (2010, methodology): “ions of strongly F-regular and sharply F-pure triples. These classes of singularities are analogous, respectively, to log terminal and log canonical singularities for complex algebraic varieties; see [HW02]. Deﬁnition 6.14. If Ris a local ring, then a triple (X= SpecR,∆,at) with ∆ ≥0 is called stron”