Rationality of peskine varieties

TL;DR

This paper investigates the rationality of Peskine sixfolds in projective space, proving rationality in specific cases and proposing a cohomological condition that predicts rationality in others, linking to hyperkähler geometry.

Contribution

It establishes the rationality of Peskine sixfolds in a particular divisor and introduces a cohomological criterion for rationality in another divisor, connecting to hyperkähler structures.

Findings

Proved rationality of Peskine sixfolds in divisor D^{3,3,10}.

Provided a cohomological condition for rationality in divisor D^{1,6,10}.

Conjectured a link between cohomological conditions and hyperkähler geometry.

Abstract

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.