Induction for secant varieties of Segre varieties

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 $p\leq 6$. We generalize a theorem of Catalisano-Geramita-Gimigliano on non-defectivity of tensor powers of $\mathbb {P}{n}$. 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 $\mathbb {P}{1}\times \mathbb {P}{1} \times \mathbb {P}{n} \times \mathbb {P}{n}$ and $\mathbb {P}{2}\times \mathbb {P}{3} \times \mathbb {P}{3}$. In the final section we propose a series of conjectures about defective Segre varieties.