Author
Kenta Hashizume
Other affiliations: Kyoto University
Bio: Kenta Hashizume is an academic researcher from University of Tokyo. The author has contributed to research in topics: Divisor (algebraic geometry) & Minimal model program. The author has an hindex of 5, co-authored 26 publications receiving 128 citations. Previous affiliations of Kenta Hashizume include Kyoto University.
Papers
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TL;DR: For a projective log canonical pair over a perfect field of characteristic larger than five, there exists a minimal model program that terminates after finitely many steps as mentioned in this paper. But this is not the case for all projective projective canonical pairs.
Abstract: Given a three-dimensional projective log canonical pair over a perfect field of characteristic larger than five, there exists a minimal model program that terminates after finitely many steps.
28 citations
TL;DR: In this paper, the minimal model theory for projective klt pairs of dimension at most n was established under the assumption that the log canonical divisor is relatively log abundant.
Abstract: Under the assumption of the minimal model theory for projective klt pairs of dimension $n$, we establish the minimal model theory for lc pairs $(X/Z,\Delta)$ such that the log canonical divisor is relatively log abundant and its restriction to any lc center has relative numerical dimension at most $n$. We also give another detailed proof of results by the second author, and study termination of log MMP with scaling.
21 citations
TL;DR: In this article, the authors studied relative log canonical pairs with relatively trivial log canonical divisors and established the minimal model theory for the pair $(X,\Delta)/Z).
Abstract: We study relative log canonical pairs with relatively trivial log canonical divisors. We fix such a pair $(X,\Delta)/Z$ and establish the minimal model theory for the pair $(X,\Delta)$ assuming the minimal model theory for all Kawamata log terminal pairs whose dimension is not greater than ${\rm dim}\,Z$. We also show the finite generation of log canonical rings for log canonical pairs of dimension five which are not of log general type.
18 citations
TL;DR: In this paper, the authors proved the boundary divisor version of the results proved by Birkar and Hacon-Xu on the relative log minimal model program (RLMM).
Abstract: We prove $$\mathbb {R}$$
-boundary divisor versions of results proved by Birkar (Publ Math Inst Hautes Etudes Sci 115(1):325–368, 2012) and Hacon–Xu (Invent Math 192(1):161–195, 2013) on special kinds of the relative log minimal model program.
17 citations
TL;DR: In this paper, a class of singularity on arbitrary pairs of a normal variety and an effective $\mathbb{R}$-divisor on it is defined, called pseudo-lc, which is a strictly extended notion of those singularities.
Abstract: We define a class of singularity on arbitrary pairs of a normal variety and an effective $\mathbb{R}$-divisor on it, which we call pseudo-lc in this paper. This is a generalization of the usual lc singularity of pairs and log canonical singularity of normal varieties introduced by de Fernex and Hacon. By giving examples of pseudo-lc pairs which are not lc or log canonical in the sense of de Fernex--Hacon's paper, we show that pseudo-lc singularity is a strictly extended notion of those singularities. We prove that pseudo-lc pairs admit a small lc modification. We also discuss a criterion of log canonicity.
12 citations
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TL;DR: In this article, the authors prove the existence of complements for pairs with DCC coefficients and the ACC for minimal log discrepancies of exceptional singularities, and show its existence for pairs of real coefficients.
Abstract: We prove the existence of $n$-complements for pairs with DCC coefficients and the ACC for minimal log discrepancies of exceptional singularities. In order to prove these results, we develop the theory of complements for real coefficients. We introduce $(n,\Gamma_0)$-decomposable $\mathbb{R}$-complements, and show its existence for pairs with DCC coefficients.
53 citations
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TL;DR: The numerical non-vanishing conjecture for projective log canonical pairs plays a key role in the minimal model program of higher dimensional algebraic geometry as discussed by the authors, and it has been proved in dimension two.
Abstract: The nonvanishing conjecture for projective log canonical pairs plays a key role in the minimal model program of higher dimensional algebraic geometry. The numerical nonvanishing conjecture considered in this paper is a weaker version of the usual nonvanishing conjecture, but valid in the more general setting of generalized log canonical pairs. We confirm it in dimension two. Under some necessary conditions we obtain effective versions of numerical nonvanishing for surfaces. Several applications are also discussed.
In higher dimensions, we mainly consider the conjecture for generalized klt pairs $(X, B+\mathbf{M})$, and reduce it to lower dimensions when $K_X+\mathbf{M}_X$ is not pseudo-effective. Up to scaling the nef part, we prove the numerical nonvanishing for pseudo-effective generalized lc threefolds with rational singularities.
33 citations
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TL;DR: In this article, the validity of the relative dlt MMP over Q-factorial three-folds in all characteristics p>0 was shown, including the rationality of klt singularities, inversion of adjunction, and normality of divisorial centres up to a universal homeomorphism.
Abstract: We show the validity of the relative dlt MMP over Q-factorial threefolds in all characteristics p>0. As a corollary, we generalise many recent results to low characteristics including: $W\mathcal{O}$-rationality of klt singularities, inversion of adjunction, and normality of divisorial centres up to a universal homeomorphism.
25 citations