Hirotachi Abo

Induction for secant varieties of Segre varieties

This paper studies the dimension of secant varieties to Segre varieties. The problem is cast both in the setting of tensor algebra and in the setting of algebraic geometry. An inductive procedure is built around the ideas of successive specializations of points and projections. This reduces the calculation of the dimension of the secant variety in a high dimensional case to a sequence of calculations of partial secant varieties in low dimensional cases. As applications of the technique: We give a complete classification of defective p-secant varieties to Segre varieties for p < 6. We generalize a theorem of Catalisano-Geramita-Gimigliano on non-defectivity of tensor powers of Pn. We determine the set of p for which unbalanced Segre varieties have defective p-secant varieties. In addition, we completely describe the dimensions of the secant varieties to the deficient Segre varieties P1 x P1 x Pn x Pn and P2 x P3 x P3. In the final section we propose a series of conjectures about defective Segre varieties.

Non-defectivity of Grassmannians of planes

Let Gr(k,n) be the Plucker embedding of the Grassmann variety of projective k-planes in Pn. For a projective variety X, lets(X) denote the variety of its s 1 secant planes. More precisely, �s(X) denotes the Zariski closure of the union of linear spans of s-tuples of points lying on X. We exhibit two functions s0(n) � s1(n) such thats(Gr(2,n)) has the expected dimension whenever n � 9 and either ss0(n) or s1(n) � s. Both s0(n) and s1(n) are asymptotic to n 2 18 . This yields, asymptotically, the typical rank of an element of ^ 3 C n+1 . Finally, we classify all defectives(Gr(k,n)) for s � 6 and provide geometric arguments underlying each defective case.

On the dimensions of secant varieties of Segre-Veronese varieties

This paper explores the dimensions of higher secant varieties to Segre-Veronese varieties. The main goal of this paper is to introduce two different inductive techniques. These techniques enable one to reduce the computation of the dimension of the secant variety in a high-dimensional case to the computation of the dimensions of secant varieties in low-dimensional cases. As an application of these inductive approaches, we will prove non-defectivity of secant varieties of certain two-factor Segre-Veronese varieties. We also use these methods to give a complete classification of defective sth Segre–Veronese varieties for small s. In the final section, we propose a conjecture about defective two-factor Segre–Veronese varieties.

Secant Varieties of Segre–Veronese Varieties ℙ m × ℙ n Embedded by O(1, 2)

Let X m,n be the Segre–Veronese variety ℙ m × ℙ n embedded by the morphism given by O(1, 2). In this paper, we provide two functions such that the sth secant variety of X m,n has the expected dimension if or . We also present a conjecturally complete list of defective secant varieties of such Segre–Veronese varieties.

On non-defectivity of certain Segre—Veronese varieties

Let Xm,n be the Segre?Veronese variety Pm×Pn embedded by the morphism given by O(1,2) and let ?s(Xm,n) denote the sth secant variety of Xm,n. In this paper, we prove that if m=n or m=n+1, then ?s(Xm,n) has the expected dimension except for ?6(X4,3). As an immediate consequence, we will give functions s1(m,n)?s2(m,n) such that if s?s1(m,n) or if s?s2(m,n), then ?s(Xm,n) has the expected dimension for all positive integers m and n.

Eigenschemes and the Jordan canonical form

Abstract We study the eigenscheme of a matrix which encodes information about the eigenvectors and generalized eigenvectors of a square matrix. The two main results in this paper are a decomposition of the eigenscheme of a matrix into primary components and the fact that this decomposition encodes the numeric data of the Jordan canonical form of the matrix. We also describe how the eigenscheme can be interpreted as the zero locus of a global section of the tangent bundle on projective space. This interpretation allows one to see eigenvectors and generalized eigenvectors of matrices from an alternative viewpoint.

Petersen plane arrangements and a surface with multiple 7-secants

We study configurations of 2-planes in P 4 that are combinatorially described by the Petersen graph. We discuss conditions for configurations to be locally Cohen-Macaulay and de- scribe the Hilbert scheme of such arrangements. An analysis of the homogeneous ideals of these configurations leads, via linkage, to a class of smooth, general type surfaces in P 4 . We compute their numerical invariants and show that they have the unusual property that they admit (multiple) 7-secants. Finally, we demonstrate that the construction applied to Petersen arrangements with additional symmetry leads to surfaces with exceptional automorphism groups.

Construction of rational surfaces of degree 12 in projective fourspace

The aim of this paper is to present two different constructions of smooth rational surfaces in projective fourspace with degree 12 and sectional genus 13. In particular, we establish the existences of five different families of smooth rational surfaces in projective fourspace with the prescribed invariants.

Implementation of Kumar's correspondence

In 1997, N.M. Kumar published a paper which introduced a new tool of use in the construction of algebraic vector bundles. Given a vector bundle on projective n-space, a well known theorem of Quillen-Suslin guarantees the existence of sections which generate the bundle on the complement of a hyperplane in projective n-space. Kumar used this fact to give a correspondence between vector bundles on projective n-space and vector bundles on projective (n−1)-space satisfying certain conditions. He then applied this correspondence to establish the existence of many, previously unknown, rank two bundles on projective fourspace in positive characteristic. The goal of the present paper is to give an explicit homological description of Kumars correspondence in a setting appropriate for implementation in a computer algebra system.