About the Defectivity of Certain Segre--Veronese Varieties We study the regularity of the higher secant varieties of $\PP^1\times \PP^n$, embedded with divisors of type $(d,2)$ and $(d,3)$. We produce, for the highest defective cases, a determinantal'' equation of the secant variety. As a corollary, we prove that the Veronese triple embedding of $\PP^n$ is not Grassmann defective. Keywords:Waring problem, Segre--Veronese embedding, secant variety, Grassmann defectivityCategories:14N15, 14N05, 14M12