Searcharxiv⌕ Search

arXiv subjects

Atsushi Ito

Publications and source records attributed to Atsushi Ito.

At least 37 records · Page 2Linked to original sources
arXiv2022

Projective normality and basepoint-freeness thresholds of general polarized abelian varieties

For a polarized abelian variety $(X,L)$, Z. Jiang and G. Pareschi introduce an invariant $β(X,L)$, called the basepoint-freeness threshold. Using this invariant, we show that a general polarized abelian variety $(X,L)$ of dimension $g$ is projectively normal if $χ(L) \geq 2^{2g-1}$ and the type of $L$ is not $(2,4,\dots,4)$. This bound is sharp since it is known that any polarized abelian variety of type $(2,4,\dots,4)$ is not projectively normal. We also give an application of $β(X,L)$ to the Infinitesimal Torelli Theorem for $Y \in |L|$.

math.AG↗
arXiv2021

M-regularity of $\mathbb{Q}$-twisted sheaves and its application to linear systems on abelian varieties

G. Pareschi and M. Popa give criterions for global generations and surjectivity of multiplication maps of global sections of coherent sheaves on abelian varieties in the theory of M-regularity. In this paper, we generalize some of their criterions via the M-regularity of $\mathbb{Q}$-twisted sheaves. As an application, we show that the M-regularity of a suitable $\mathbb{Q}$-twisted sheaf implies property $(N_p)$ and jet-ampleness for ample line bundles on abelian varieties.

math.AG↗
arXiv2020

Successive minima of line bundles

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 toric varieties, the successive minima are equivalent to the (reciprocal of) usual successive minima of the difference of the moment polytope.

math.AG↗
arXiv2019

A combinatorial description of dual defects of toric varieties

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.

math.AG↗
arXiv2019

Projective reconstruction in algebraic vision

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 [HS09].

math.AG↗
arXiv2019

Derived equivalence and Grothendieck ring of varieties: the case of K3 surfaces of degree 12 and abelian varieties

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 $\mathbb{L} = [ \mathbb{A}^1 ]$ 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 $\mathbb{L}$, thereby giving a negative answer to the problem. We also discuss variations of the problem.

math.AG↗
arXiv2019

Complete intersection Calabi--Yau manifolds with respect to homogeneous vector bundles on Grassmannians

Based on the method by [Küc95], 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.

math.AG↗
arXiv2018

Quasi-free-standing monolayer hexagonal boron nitride on Ni

The electronic structure of monolayer hexagonal boron nitride grown on Ni by the diffusion and precipitation method was studied by x-ray absorption spectroscopy, emission spectroscopy, x-ray photoelectron spectroscopy and micro-ultraviolet photoemission spectroscopy. No indication of hybridization between h-BN pi and Ni 3d orbitals was observed. That is, the monolayer h-BN was found to be in the quasi-free-standing state. These results are in striking contrast to those of previous studies in which h-BN was strongly bound to the Ni surface by the orbital hybridization. The absence of hybridization is attributed to absence of a Ni(111) surface in this study. The lattice-matched Ni(111) surface is considered to be essential to orbital hybridization between h-BN and Ni.

cond-mat.mes-hall↗
arXiv2017

On separable higher Gauss maps

We study the $m$-th Gauss map in the sense of F.~L.~Zak of a projective variety $X \subset \mathbb{P}^N$ over an algebraically closed field in any characteristic. For all integer $m$ with $n:=\dim(X) \leq m < N$, we show that the contact locus on $X$ of a general tangent $m$-plane is a linear variety if the $m$-th 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 $\mathbb{P}^1 \times \mathbb{P}^n \subset \mathbb{P}^{2n-1}$. This is related to L. Ein's classification of varieties with small dual varieties in characteristic zero.

math.AG↗
arXiv2015

On Gauss maps in positive characteristic in view of images, fibers, and field extensions

The Gauss map of a projective variety $X \subset \mathbb{P}^N$ 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 $X$.

math.AG↗
arXiv2014

On birational geometry of the space of parametrized rational curves in Grassmannians

In this paper, we study the birational geometry of the Quot schemes of trivial bundles on $\mathbb{P}^1$ by constructing small $\mathbb{Q}$-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.

math.AG↗
arXiv2014

Examples of Mori dream spaces with Picard number two

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.

math.AG↗