Peskine sixfolds and Debarre-Voisin fourfolds with associated cubic fourfolds

TL;DR

The paper develops a period-map–driven framework to study when Peskine sixfolds $X_1^\sigma$ and Debarre--Voisin fourfolds $X_6^\sigma$ admit Hodge-theoretic associations to K3 surfaces or cubic fourfolds, encoded by discriminant divisors $\mathcal{P}_d$. It proves arithmetic criteria on the discriminant $d$ characterizing such associations and, in discriminant $24$, shows that the associated cubic fourfold $Y$ has a Fano variety of lines isomorphic to $X_6^\sigma$, yielding a concrete geometric bridge between the two families. The work connects lattice-theoretic data, period domains, and geometric constructions (Palatini, Hassett divisors) to classify special Peskine sixfolds and their associated K3s and cubic fourfolds, offering explicit examples and implications for birational geometry and rationality through twisted K3 structures. Overall, the results illuminate how K3-type hyperkähler geometry and cubic fourfolds intertwine with Peskine sixfolds via precise discriminant and marking conditions, enriching the tapestry of Fano and hyperkähler correspondences.

Abstract

We develop the notion of Peskine sixfolds with associated K3 surfaces and cubic fourfolds and work out numerical conditions for when these associations occur. In discriminant 24, the first family for which there is an associated cubic fourfold, we identify the cubic explicitly. Moreover, we prove that in this case the Fano variety of lines of the cubic fourfold is isomorphic to the associated Debarre-Voisin hyperkähler fourfold.

Theorems & Definitions (36)

• Theorem 1.1
• Theorem 1.2
• Remark 2.1
• Theorem 2.2
• Definition 3.1
• Remark 3.2
• Definition 3.3
• Definition 3.4
• Remark 3.5
• Definition 3.6