arXiv subjects

Katsuhisa Furukawa

Publications and source records attributed to Katsuhisa Furukawa.

12 recordsLinked to original sources

arXiv2026

On the prime ideals of higher secant varieties of Veronese embeddings of small degrees

In this paper, we study minimal generators of the (saturated) defining ideal of the $k$-secant variety $σ_k(v_d(\mathbb{P}^n))$ of the image of the $d$-uple Veronese embedding $v_d: \mathbb{P}^n \rightarrow \mathbb{P}^N$ with ${N=\binom{n+d}{d}-1}$, focusing on cases where the degree of $σ_k(v_d(\mathbb{P}^n))$ is relatively small. First, we show that the prime ideal $I(σ_4(v_3(\mathbb{P}^3)))$ is minimally generated by $36$ homogeneous polynomials of degree $5$. This implies that $σ_4(v_3(\mathbb{P}^3)) \subset \mathbb{P}^{19}$ is a del Pezzo $4$-secant variety (i.e., $\mathrm{deg}(σ_4(v_3(\mathbb{P}^3))) = 105$ and the sectional genus $π(σ_4(v_3(\mathbb{P}^3))) = 316$), thereby providing a new example of an arithmetically Gorenstein variety of codimension $4$. This result addresses the symmetric version of the ``Salmon problem'' posed by E. Allman in \cite{Allman}. As an application, we decide the non-singularity of a certain locus in $σ_4(v_3(\mathbb{P}^3))$. Furthermore, by inheritance, we obtain the generators of $I(σ_4(v_3(\mathbb{P}^n)))$ for all $n \geq 3$. Based on the method used for $σ_4(v_3(\mathbb{P}^3))$, we also propose a procedure to compute the first non-trivial degree piece, $I(σ_k(v_d(\mathbb{P}^n)))_{k+1}$, for the general $k$-secant case using prolongation and weight space decomposition. Applying this procedure, we present a few more cases of $k$-secant varieties of relatively small degrees; in each of these cases, the ideal is generated in degree $k+1$ and can be fully determined by explicitly computing all generators within this degree piece.

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arXiv2025

On the singular loci of higher secant varieties of Veronese embeddings

The $k$-th secant variety of a projective variety $X \subset \mathbb{P}^N$, denoted by $σ_k(X)$, is defined to be the closure of the union of $(k-1)$-planes spanned by $k$ points on $X$. In this paper, we examine the $k$-th secant variety $σ_k(v_d(\mathbb{P}^n)) \subset \mathbb{P}^N$ of the image of the $d$-uple Veronese embedding $v_d$ of $\mathbb{P}^n$ to $\mathbb{P}^N$ with $N=\binom{n+d}{d}-1$, and focus on the singular locus of $σ_k(v_d(\mathbb{P}^n))$, which is only known for $k\le3$. To study the singularity for arbitrary $k,d,n$, we define \emph{the $m$-subsecant locus} of $σ_k(v_d(\mathbb{P}^n))$ to be the union of $σ_k(v_d(\mathbb{P}^m))$ with any $m$-plane $\mathbb{P}^m \subset \mathbb{P}^n$. By investigating the projective geometry of moving embedded tangent spaces along subvarieties and using known results on the secant defectivity and the identifiability of symmetric tensors, we determine whether the $m$-subsecant locus is contained in the singular locus of $σ_k(v_d(\mathbb{P}^n))$ or not. Depending on the value of $k$, these subsecant loci show an interesting trichotomy between generic smoothness, non-trivial singularity, and trivial singularity. In many cases, they can be used as a new source for the singularity of the $k$-th secant variety of $v_d(\mathbb{P}^n)$ other than the trivial one, the $(k-1)$-th secant variety of $v_d(\mathbb{P}^n)$. We also consider the case of the $4$-th secant variety of $v_d(\mathbb{P}^n)$ by applying main results and computing conormal space via a certain type of Young flattening. Finally, we present some generalizations and discussions for further developments.

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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.

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arXiv2018

Cubic hypersurfaces with positive dual defects

We show that if a cubic hypersurface with positive dual defect over the complex number field is not a cone, then either the hypersurface coincides with the secant variety of the singular locus, or the hypersurface contains a linear subvariety of dimension greater than the dual defect such that the intersection of the singular locus and a general contact locus is contained in the linear subvariety.

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arXiv2017

Dimension of the space of conics on Fano hypersurfaces

R. Beheshti showed that, for a smooth Fano hypersurface $X$ of degree $\leq 8$ over the complex number field $\mathbb{C}$, the dimension of the space of lines lying in $X$ is equal to the expected dimension. We study the space of conics on $X$. In this case, if $X$ contains some linear subvariety, then the dimension of the space can be larger than the expected dimension. In this paper, we show that, for a smooth Fano hypersurface $X$ of degree $\leq 6$ over $\mathbb{C}$, and for an irreducible component $R$ of the space of conics lying in $X$, if the $2$-plane spanned by a general conic of $R$ is not contained in $X$, then the dimension of $R$ is equal to the expected dimension.

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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.

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arXiv2015

On general fibers of Gauss maps in positive characteristic

A general fiber of the Gauss map of a projective variety in $\mathbb{P}^N$ coincides with a linear subvariety of $\mathbb{P}^N$ in characteristic zero. In positive characteristic, S. Fukasawa showed that a general fiber of the Gauss map can be a non-linear variety. In this paper, we show that each irreducible component of such a possibly non-linear fiber of the Gauss map is contracted to one point by the degeneracy map, and is contained in a linear subvariety corresponding to the kernel of the differential of the Gauss map. We also show the inseparability of Gauss maps of strange varieties not being cones.

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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 $X$.

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arXiv2014

Convex separably rationally connected complete intersections

We give a generalization of a result of R. Pandharipande to arbitrary characteristic: We prove that, if $X$ is a convex, separably rationally connected, smooth complete intersection in $\mathbb{P}^N$ over an algebraically closed field of arbitrary characteristic, then $X$ is rational homogeneous.

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arXiv2014

Gauss maps of toric varieties

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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arXiv2014

Cohomological characterization of hyperquadrics of odd dimensions in characteristic two

We consider characterizations of projective varieties in terms of their tangents. S. Mori established the characterization of projective spaces in arbitrary characteristic by ampleness of tangent bundles. J. Wahl characterized projective spaces in characteristic zero by cohomological condition of tangent bundles; in addition, he remarked that a counter-example in characteristic two is constructed from odd-dimensional hyperquadrics $Q_{2n-1}$ with $n > 1$. This is caused by existence of a common point in $\mathbb{P}^{2n}$ which every embedded tangent space to the quadric contains. In general, a projective variety in $\mathbb{P}^N$ is said to be strange if its embedded tangent spaces admit such a common point in $\mathbb{P}^N$. A non-linear smooth projective curve is strange if and only if it is a conic in characteristic two (E. Lluis, P. Samuel). S. Kleiman and R. Piene showed that a non-linear smooth hypersurface in $\mathbb{P}^N$ is strange if and only if it is a quadric of odd-dimension in characteristic two. In this paper, we investigate complete intersection varieties, and prove that, a non-linear smooth complete intersection variety in $\mathbb{P}^N$ is strange if and only if it is a quadric in $\mathbb{P}^N$ of odd dimension in characteristic two; these conditions are also equivalent to non-vanishing of 0-cohomology of (-1)-twist of the tangent bundle.

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arXiv2013

Duality with expanding maps and shrinking maps, and its applications to Gauss maps

We study expanding maps and shrinking maps of subvarieties of Grassmann varieties in arbitrary characteristic. The shrinking map was studied independently by Landsberg and Piontkowski in order to characterize Gauss images. To develop their method, we introduce the expanding map, which is a dual notion of the shrinking map and is a generalization of the Gauss map. Then we give a characterization of separable Gauss maps and their images, which yields results for the following topics: (1) Linearity of general fibers of separable Gauss maps; (2) Generalization of the characterization of Gauss images; (3) Duality on one-dimensional parameter spaces of linear subvarieties lying in developable varieties.

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