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