On the dimensions of secant varieties of Segre-Veronese varieties

TL;DR

This paper introduces inductive techniques to determine the dimensions of secant varieties of Segre-Veronese varieties, enabling reduction from high to low dimensions, and classifies defectivity for small secant orders.

Contribution

It develops two new inductive methods for computing secant variety dimensions and applies them to classify defectivity in two-factor Segre-Veronese varieties.

Findings

Proved non-defectivity for certain two-factor Segre-Veronese varieties.

Provided a complete classification of defective varieties for small secant orders.

Proposed a conjecture on defectivity of two-factor Segre-Veronese varieties.

Abstract

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 s-th Segre-Veronese varieties for small s. In the final section, we propose a conjecture about defective two-factor Segre-Veronese varieties.