Segre-Veronese embeddings of $\mathbb{P}^1\times \mathbb{P}^1\times \mathbb{P}^1$ and their secant varieties

In this paper we compute the dimension of all the $s^{th}$ higher secant varieties of the Segre-Veronese embeddings $Y_{\underline d}$ of the product $\mathbb{P}^1\times \mathbb{P}^1\times \mathbb{P}^1$ in the projective space $\mathbb{P}^N$ via divisors of multi-degree $\underline d = (a, b, c) (N = (a + 1)(b+1)(c+1)-1)$. We find that $Y_{\underline d}$ has no deficient higher secant varieties, unless $\underline d$ = (2, 2, 2) and $s$ = 7, or $\underline d$ = (2h, 1, 1) and $s$ = 2h + 1, with defect 1 in both cases.