On the Secant Defectivity of Segre-veronese Varieties

Let X ⊆ PN be a non-degenerate projective variety of dimension d. Then the sth secant variety of X, denoted σs(X), is the Zariski closure of the union of linear spans of s-tuples of points lying on X. The study of secant varieties has a long history. The interest in this subject goes back to the Italian school at the turn of the 20th century. This topic has received renewed interest over the past several decades, mainly due to its increasing importance in an ever widening collection of disciplines including algebraic complexity theory [13, 26, 27], algebraic statistics [23, 22, 8], and combinatorics [29, 30]. The major questions surrounding secant varieties center around finding invariants of those objects such as dimension. A simple dimension count suggests that the expected dimension of σs(X) is min{s(d + 1) − 1, N}. We say that X has a defective sth secant variety if σs(X) does not have the expected dimension. In particular, X is said to be defective if X has a defective sth secant variety for some s. The paper explores problems related to the classification of defective secant varieties of Sege-Veronese varieties. This is analogous to the celebrated theorem of Alexander and Hirschowitz [7], which asserts that higher secant varieties of Veronese varieties have the expected dimension (modulo a fully described list of exceptions). This work completed the Waring problem for polynomials which had remained unsolved for some time. There are corresponding conjecturally complete lists of defective secant varieties for Segre varieties [5] and for Grassmann varieties [25, 10]. Secant varieties of Segre-Veronese varieties are however less well-understood. In recent years, considerable efforts have been made to develop techniques to study secant varieties of Segre-Veronese varieties (see for example [18, 15, 14, 28, 18, 9, 16, 6]). But even the classification of defective two-factor Segre-Veronese varieties is still far from complete. Very recently, the non-defectivity of two-factor Segre-Veronse varieties was systematically studied in [2], where the authors suggested that secant varieties of such Segre-Veronese varieties are not defective modulo the list of the well known exceptions. More precisely, they proposed the following conjecture: