Secant varieties of Segre-Veronese varieties P^m x P^n embedded by O(1,2)

Let $X_{m,n}$ be the Segre-Veronese variety $$\mathbb{P}$^m \times $\mathbb{P}$^n$embeddedbythemorphismgivenby$$\mathcal{O}$(1,2)$. In this paper, we provide two functions $\underset{‾}{s}(m,n)\le \bar{s}(m,n)$ such that the $s^{\mathrm{t}\mathrm{h}}$ secant variety of $X_{m,n}$ has the expected dimension if $s\le \underset{‾}{s}(m,n)$ or $\bar{s}(m,n)\le s$. We also present a conjecturally complete list of defective secant varieties of such Segre-Veronese varieties.