Summary: ON THE DIMENSIONS OF SECANT VARIETIES OF
HIROTACHI ABO AND MARIA CHIARA BRAMBILLA
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 sth Segre-
Veronese varieties for small s. In the final section, we propose a conjecture
about defective two-factor Segre-Veronese varieties.
In many applications, it is natural to represent a collection of data as a multi-
indexed list. Alternatively, one can think of the data as a multi-dimensional array.
A mathematical framework that includes the study of multi-dimensional arrays is
through parameter spaces of tensors.
Every tensor can be written as a linear combination of so-called decomposable
tensors. A tensor is said to have rank s if it can be written as a linear combination