Existence of Mori fibre spaces for 3-folds in char $p$

Birkar, Caucher; Waldron, Joe

doi:10.48550/arxiv.1410.4511

Cited by 9 publications

(17 citation statements)

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“…We will work over an algebraically closed field k of characteristic p > 5.. The existence of good log minimal models for klt threefold pairs in characteristic p > 5 is proven in [2] when K X + B is big and generalised in [3] to the case where only B is big. For klt threefold pairs with arbitrary boundary this would follow from [2] and the log abundance conjecture.…”

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confidence: 99%

Finite generation of the $\log$ canonical ring for $3$-folds in char $p$

Waldron

2017

Mathematical Research Letters

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We prove that the log canonical ring of a projective klt pair of dimension 3 with Q-boundary over an algebraically closed field of characteristic p > 5 is finitely generated. In the process we prove log abundance for such pairs in the case κ = 2.Proof. [15, A018] Lemma 2.2 (Zariski's Main Theorem I). Let f : X → Y be a quasi-finite and separated map of algebraic spaces over a scheme S, such that Y is quasicompact and quasi-separated. Then there exists a factorisation X → T → Y so that X → T is a quasi-compact open immersion and T → Y is finite.

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confidence: 99%

Finite generation of the $\log$ canonical ring for $3$-folds in char $p$

Waldron

2017

Mathematical Research Letters

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“…By the characteristic assumption, the ring S is F -pure. Set R to be the k-subalgebra generated by the monomials w 4 , w 3 x, w 2 x 2 , wx 3 , x 4 , y 4 , y 3 z, y 2 z 2 , yz 3 , z 4 .…”

Section: Positive Characteristic Case Imentioning

confidence: 99%

“…The ring R has a standard grading under which each of the monomials displayed is assigned degree one. Computing the socle modulo the system of parameters x 4 , y 4 , z 4 , it follows that a(R) = −1. By Proposition 3.6, we have c(m) = 1.…”

Section: Positive Characteristic Case Imentioning

confidence: 99%

“…that Y is a normal projective variety of Fano type of dimension at most three. It is known that the minimal model program holds for klt surfaces and also for three-dimensional klt pairs in characteristic p > 5 (see [15,2,4]). Thus, applying essentially the same argument as the proof of [3, Corollary 1.1.9], we can see that…”

Section: Positive Characteristic Case Imentioning

confidence: 99%

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A Gorenstein Criterion for StronglyF-Regular and Log Terminal Singularities

Singh

Takagi

Varbaro

2016

Int Math Res Notices

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Dedicated to Professor Craig Huneke on the occasion of his sixty-fifth birthday.Abstract. A conjecture of Hirose, Watanabe, and Yoshida offers a characterization of when a standard graded strongly F -regular ring is Gorenstein, in terms of an F -pure threshold. We prove this conjecture under the additional hypothesis that the anti-canonical cover of the ring is Noetherian. Moreover, under this hypothesis on the anti-canonical cover, we give a similar criterion for when a normal F -pure (resp. log canonical) singularity is quasi-Gorenstein, in terms of an F -pure (resp. log canonical) threshold.

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“…Recently, starting from the work of Hacon and Xu [HX15], many of the classical results of the minimal model programme in characteristic zero have been extended to three dimensional varieties over an algebraically closed field k of characteristic p > 5 [Bir16, CTX15,BW14].…”

Section: Introductionmentioning

confidence: 99%