Secant Varieties of Segre-Veronese Varieties $\P^m \times \P^n$ Embedded by $\cO(1,2)$

Abstract

Let $X_{m,n}$ be the Segre-Veronese variety $\P^m \times \P^n$ embedded by the morphism given by $\cO(1,2)$. In this paper, we provide two functions $\underline{s}(m,n)\le \overline{s}(m,n)$ such that the $s$th secant variety of $X_{m,n}$ has the expected dimension if $s \leq \underline{s}(m,n)$ or $ \overline{s}(m,n) \leq s$. We also present a conjecturally complete list of defective secant varieties of such Segre-Veronese varieties.