Rationality of Peskine varieties

We study the rationality of the Peskine sixfolds in \({\textbf{P}}^9\). We prove the rationality of the Peskine sixfolds in the divisor \({\mathcal {D}}^{3,3,10}\) inside the moduli space of Peskine sixfolds and we provide a cohomological condition which ensures the rationality of the Peskine sixfolds in the divisor \({\mathcal {D}}^{1,6,10}\) [(notation from Benedetti and Song (Divisors in the moduli space of Debarre-Voisin varieties, http://arxiv.org/abs/2106.06859, 2021)]. We conjecture, as in the case of cubic fourfolds containing a plane, that the cohomological condition translates into a cohomological and geometric condition involving the Debarre-Voisin hyperkähler fourfold associated to the Peskine sixfold.