Remarks on the geometry of the variety of planes of a cubic fivefold

TL;DR

This paper investigates the geometric properties of the variety of planes on a cubic fivefold, including its embedding via the Gauss map and its relation to osculating planes.

Contribution

It derives a cotangent bundle sequence for the variety of planes and proves the Gauss map is an embedding, linking it to the variety of lines on a cubic fourfold.

Findings

Gauss map of the variety of planes is an embedding.

The variety of planes is a Lagrangian subvariety of the variety of lines.

Relation established between osculating planes of a cubic fourfold and planes of a cyclic cubic fivefold.

Abstract

This note presents some properties of the variety of planes $F_2(X)\subset G(3,7)$ of a cubic $5$-fold $X\subset \mathbb P^6$. A cotangent bundle exact sequence is first derived from the remark made by Iliev and Manivel that $F_2(X)$ sits as a Lagrangian subvariety of the variety of lines of a cubic $4$-fold, which is a hyperplane section of $X$. Using the sequence, the Gauss map of $F_2(X)$ is then proven to be an embedding. The last section is devoted to the relation between the variety of osculating planes of a cubic $4$-fold and the variety of planes of the associated cyclic cubic $5$-fold.