Cubic fourfolds containing a plane and K3 surfaces of Picard rank two

TL;DR

This paper investigates specific cubic fourfolds in complex projective space containing a plane, focusing on their associated quadric bundles that lack rational sections, revealing new geometric properties.

Contribution

It introduces families of cubic fourfolds with planes where the related quadric bundles do not admit rational sections, highlighting novel geometric phenomena.

Findings

Identification of families of cubic fourfolds with special geometric properties

Demonstration of quadric bundles without rational sections

Insights into the structure of K3 surfaces related to these fourfolds

Abstract

We present some families of cubic hypersurfaces in $\mathbb P^5 (\mathbb C)$ containing a plane whose associated quadric bundle does not have a rational section.