Lines on cubic threefolds and fourfolds containing a plane

TL;DR

This paper studies the geometry of lines on cubic threefolds and fourfolds containing a plane, using Fano schemes to analyze rationality and construct explicit Lagrangian fibrations.

Contribution

It provides a detailed description of the Fano scheme of lines on such cubic threefolds and fourfolds, and applies this to rationality criteria and explicit fibrations.

Findings

Characterization of the Fano scheme of lines on cubic threefolds with a plane

Criteria for rationality of these cubic threefolds over nonclosed fields

Explicit construction of a Lagrangian fibration on the Fano variety of lines on a cubic fourfold

Abstract

We describe the Fano scheme of lines on a general cubic threefold containing a plane over a field $k$ of characteristic different from 2. Then, we use the Fano scheme to characterize rationality for such cubic threefolds over nonclosed fields and to construct a Lagrangian fibration from the Fano variety of lines on a cubic fourfold containing a plane explicitly.