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 \(\mathcal {L}\), the paper establishes, under certain conditions, the regularity of \((Y \times {\mathbb {P}}^2,\mathcal {L}[t])\) for \(t\ge 2\). The mildness of those conditions allow to classify all secant defective cases of any product of \(({\mathbb {P}}^1)^{ j}\times ({\mathbb {P}}^2)^{k}\), \(j,k \ge 0\), embedded in multidegree at least \((2, \ldots , 2)\) and \((\mathbb {P}^m\times \mathbb {P}^n\times (\mathbb {P}^2)^k, \mathcal {O}_{\mathbb {P}^m\times \mathbb {P}^n\times (\mathbb {P}^2)^k} (d,e,t_1, \ldots , t_k))\) where \(d,e \ge 3\), \(t_i\ge 2\), for any n and m.