Cubic fourfolds containing a plane and a quintic del Pezzo surface

TL;DR

This paper studies special smooth rational cubic fourfolds containing a plane and a quintic del Pezzo surface, revealing their geometric properties, Brauer classes, and implications for derived categories and rationality conjectures.

Contribution

It identifies a specific class of cubic fourfolds with nontrivial Brauer classes and demonstrates their geometric and categorical properties, supporting Kuznetsov's rationality conjecture.

Findings

Existence of cubic fourfolds with nontrivial Brauer class

Connection between geometric configurations and derived equivalences

Identification of irreducible components in the moduli space

Abstract

We isolate a class of smooth rational cubic fourfolds X containing a plane whose associated quadric surface bundle does not have a rational section. This is equivalent to the nontriviality of the Brauer class of the even Clifford algebra over the K3 surface S of degree 2 arising from X. Specifically, we show that in the moduli space of cubic fourfolds, the intersection of divisors C_8 and C_14 has five irreducible components. In the component corresponding to the existence of a tangent conic to the sextic degeneration curve of the quadric bundle, we prove that the general member is both pfaffian and has nontrivial Brauer class. Such cubic fourfolds also provide twisted derived equivalences between K3 surfaces of degree 2 and 14, hence further corroboration of Kuznetsov's derived categorical conjecture on the rationality of cubic fourfolds.