  We study the rationality of the Peskine sixfolds in P^9. We prove the
rationality of the Peskine sixfolds in the divisor 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 D^{1,6,10}
(notation from [BS]). 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
hyperk{\"a}hler fourfold associated to the Peskine sixfold.
