Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER MATHEMATICAL SOC  

G. Pareschi and M. Popa [J. Amer.Math. Soc. 16 (2003), pp. 285-302] give criterions for global generation and surjectivity of multiplication maps of global sections of coherent sheaves on abelian varieties in the theory of M-regularity. In this paper, we refine some of their criterions via the M-regularity of Q-twisted sheaves introduced by Z. Jiang and G. Pareschi [Ann. Sci. Ec. Norm. Super. (4) 53 (2020), pp. 815-846]. As an application, we show that the M-regularity of a suitable Q-twisted sheaf implies property (N-p) and jet-ampleness for ample line bundles on abelian varieties.

DOI： 10.1090/tran/8725

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Cellule MathDoc/CEDRAM  

It is known that the automorphism group of a K-polystable Fano manifold is reductive. Codogni and Dervan constructed a canonical filtration of the section ring, called Loewy filtration, and conjectured that the filtration destabilizes any Fano variety with non-reductive automorphism group. In this note, we give a counterexample to their conjecture.

DOI： 10.5802/aif.3395

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WORLD SCIENTIFIC PUBL CO PTE LTD  

From a finite set in a lattice, we can define a toric variety embedded in a projective space. In this paper, we give a combinatorial description of the dual defect of the toric variety using the structure of the finite set as a Cayley sum with suitable conditions. We also interpret the description geometrically.

DOI： 10.1142/S0219199720500017

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：CAMBRIDGE UNIV PRESS  

We discuss the geometry of rational maps from a projective space of an arbitrary dimension to the product of projective spaces of lower dimensions induced by linear projections. In particular, we give an algebro-geometric variant of the projective reconstruction theorem by Hartley and Schaffalitzky.

DOI： 10.4153/S0008439519000687

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER INTERNATIONAL PUBLISHING AG  

In this paper, we discuss the problem of whether the difference [ X]-[Y] of the classes of a Fourier-Mukai pair (X, Y) of smooth projective varieties in the Grothendieck ring of varieties is annihilated by some power of the class L = [A(1)] of the affine line. We give an affirmative answer for Fourier-Mukai pairs of very general K3 surfaces of degree 12. On the other hand, we prove that in each dimension greater than one, there exists an abelian variety such that the difference with its dual is not annihilated by any power of L, thereby giving a negative answer to the problem. We also discuss variations of the problem.

DOI： 10.1007/s00029-020-00561-x

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

We introduce and study the successive minima of line bundles on proper algebraic varieties. The first (resp. last) minima are the width (resp. Seshadri constant) of the line bundle at very general points. The volume of the line bundle is equivalent to the product of the successive minima. For line bundles on tonic varieties, the successive minima are equivalent to the (reciprocal of) usual successive minima of the difference of the moment polytope. (C) 2020 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.aim.2020.107045

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER HEIDELBERG  

Based on the method by Kuchle (Math Z 218(4), 563-575, 1995), we give a procedure to list up all complete intersection Calabi-Yau manifolds with respect to direct sums of irreducible homogeneous vector bundles on Grassmannians for each dimension. In particular, we give a classification of such Calabi-Yau 3-folds and determine their topological invariants. We also give alternative descriptions for some of them.

DOI： 10.1007/s00209-018-2163-5

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：MICHIGAN MATHEMATICAL JOURNAL  

We study the mth Gauss map in the sense of F. L. Zak of a projective variety X subset of P-N over an algebraically closed field in any characteristic. For all integers m with n := dim(X) <= m < N, we show that the contact locus on X of a general tangent m-plane is a linear variety if the mth Gauss map is separable. We also show that for smooth X with n < N - 2, the (n + 1)th Gauss map is birational if it is separable, unless X is the Segre embedding P-1 x P-n subset of P2n-1.This is related to Ein's classification of varieties with small dual varieties in characteristic zero.

DOI： 10.1307/mmj/1555574416

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：UNIV PRESS INC  

Motivated by [J. Algebraic Geom. 27 (2018), pp. 203-209] and [C. R. Math. Acad. Sci. Paris 354 (2016), pp. 936-939], we show the equality ([X] - [Y]) . [A( l) ] = 0 in the Grothendieck ring of varieties, where (X, Y) is a pair of Calabi-Yau 3-folds cut out from the pair of Grassmannians of type G(2).

DOI： 10.1090/jag/731

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：TAYLOR & FRANCIS INC  

In this note, we give a slight improvement of a result of A. Kuronya and V. Lozovanu about higher syzygies on abelian surfaces.

DOI： 10.1080/00927872.2018.1464172

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER MATHEMATICAL SOC  

In this paper, we study the birational geometry of the Quot schemes of trivial bundles on P-1 by constructing small Q-factorial modifications of the Quot schemes as suitable moduli spaces. We determine all the models which appear in the minimal model program on the Quot schemes. As a corollary, we show that the Quot schemes are Mori dream spaces and log Fano.

DOI： 10.1090/tran/6840

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：INT PRESS BOSTON, INC  

We study I-functions of Calabi-Yau 3-folds with Picard number one which are zero loci of general sections of direct sums of globally generated irreducible homogeneous vector bundles on Grassmannians.

DOI： 10.4310/CNTP.2017.v11.n2.a2

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：OXFORD UNIV PRESS  

The Gauss map of a projective variety X. PN is a rational map from X to a Grassmann variety. In positive characteristic, we show the following results. (1) For given projective varieties F and Y, we construct a projective variety X whose Gauss map has F as its general fiber and has Y as its image. More generally, we give such construction for families of varieties over Y instead of fixed F. (2) At least in the case when the characteristic is not equal to 2, any inseparable field extension appears as the extension induced from the Gauss map of some X.

DOI： 10.1093/imrn/rnw080

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：TOHOKU UNIVERSITY  

We investigate Gauss maps of (not necessarily normal) projective toric varieties over an algebraically closed field of arbitrary characteristic. The main results are as follows: (1) The structure of the Gauss map of a toric variety is described in terms of combinatorics in any characteristic. (2) We give a developability criterion in the toric case. In particular, we show that any toric variety whose Gauss map is degenerate must be the join of some toric varieties in characteristic zero. (3) As applications, we provide two constructions of toric varieties whose Gauss maps have some given data (e.g., fibers, images) in positive characteristic.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

In this paper, we give an algebro-geometric characterization of Cayley polytopes. As a special case, we also characterize lattice polytopes with lattice width one by using Seshadri constants. (C) 2014 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.aim.2014.11.010

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ANNALES INST FOURIER  

We construct smooth irreducible curves of the lowest possible degrees in quadric and cubic surfaces whose complements are Kobayashi hyperbolically imbedded into those surfaces. Moreover we characterize line bundles on quadric and cubic surfaces such that the complete linear systems of the line bundles have a smooth irreducible curve whose complement is Kobayashi hyperbolically imbedded into those surfaces.

DOI： 10.5802/aif.2982

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER HEIDELBERG  

In this note, we give a sufficient condition such that a projective variety with Picard number two is a Mori dream space. Using this condition, we obtain examples of Mori dream spaces with Picard number two.

DOI： 10.1007/s00229-014-0673-y

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WALTER DE GRUYTER GMBH  

We give a method to estimate Seshadri constants on toric varieties at any point. By using the estimations and toric degenerations, we can obtain some new computations or estimations of Seshadri constants on non-toric varieties. In particular, we investigate Seshadri constants on hypersurfaces in projective spaces and Fano 3-folds with Picard number 1 in detail.

DOI： 10.1515/crelle-2012-0116

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER HEIDELBERG  

In this paper, we study a relation between Seshadri constants and degrees of defining polynomials. In particular, we compute the Seshadri constants on Fano varieties obtained as complete intersections in rational homogeneous spaces of Picard number one.

DOI： 10.1007/s00208-013-0969-3

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

Okounkov bodies, which are closed convex sets defined for big line bundles, have rich information on the line bundles. On the other hand, Seshadri constants are invariants which measure the positivity of line bundles. In this paper, we prove that Okounkov bodies give lower bounds of Seshadri constants. (C) 2013 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.aim.2013.04.005

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Language：English   Publishing type：Article, review, commentary, editorial, etc. (scientific journal)  

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