Induction for secant varieties of Segre varieties

Abstract

Abstract. 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 × P1 × Pn × Pn and P2 × P3 × P3. In the final section we propose a series of conjectures about defective Segre varieties.

Tensors : A brief introduction

“…As in the case of 2nd order moments, it is a contravariant tensor. In fact, if x ′ = Ax, y ′ = By and z ′ = Cz, then M ′ = (A, B, C) · M as in (1). It turns out that cumulants may also be seen as tensors as pointed out in [57].…”

“…However, things are not so simple, as early pointed out by Clebsch in the nineteenth century. In fact, there are exceptions [3], [1], [23], [58].…”

Geometry and the complexity of matrix multiplication

“…The most interesting exception in the case of secant varieties of Segre varieties is σ 4 (Seg(P 2 × P 2 × P 2 )), which is discussed in detail in §6. In the proof of Lemma 3.16 of [1], a geometric explanation of the degeneracy is given: any four points on Seg(…”

“…These papers use methods similar to those of Strassen and Lickteig, but the language is more geometric (fat points, degeneration arguments). Some explanation of the relation between the algebreo-geometric and tensor language is given in [1].…”

Dimension of Marginals of Kronecker Product Models

“…The first line is (2). The second line follows from the continuity of the parametrization of the exponential family (see Definition 2) and the lower semicontinuity of the rank function.…”