Tensoring by a plane maintains secant-regularity in degree at least two

Starting from an integral projective variety Y equipped with a very ample, non-special and not-secant defective line bundle \(\mathcal {L}\) $\mathcal{L}$ , the paper establishes, under certain conditions, the regularity of \((Y \times {\mathbb {P}}^2,\mathcal {L}[t])\) $(Y\times {\mathbb{P}}^2,\mathcal{L}[t])$ for \(t\ge 2\) $t\ge 2$ . The mildness of those conditions allow to classify all secant defective cases of any product of \(({\mathbb {P}}^1)^{ j}\times ({\mathbb {P}}^2)^{k}\) ${({\mathbb{P}}^1)}^j\times {({\mathbb{P}}^2)}^k$ , \(j,k \ge 0\) $j,k\ge 0$ , embedded in multidegree at least \((2, \ldots , 2)\) $(2,\dots ,2)$ and \((\mathbb {P}^m\times \mathbb {P}^n\times (\mathbb {P}^2)^k, \mathcal {O}_{\mathbb {P}^m\times \mathbb {P}^n\times (\mathbb {P}^2)^k} (d,e,t_1, \ldots , t_k))\) $({\mathbb{P}}^m\times {\mathbb{P}}^n\times {({\mathbb{P}}^2)}^k,{\mathcal{O}}_{{\mathbb{P}}^m\times {\mathbb{P}}^n\times {({\mathbb{P}}^2)}^k}(d,e,t_1,\dots ,t_k))$ where \(d,e \ge 3\) $d,e\ge 3$ , \(t_i\ge 2\) $t_i\ge 2$ , for any n and m.