Eight cubes of linear forms in $\p^6$

TL;DR

This paper provides a geometric explanation for the Weak Lefschetz Property (WLP) of an ideal generated by eight cubes of linear forms in projective six-space, highlighting degrees where WLP holds or fails.

Contribution

It offers a geometric insight into the WLP behavior of a specific ideal in projective space, clarifying why it holds in degree 3 and fails in degree 5.

Findings

The ideal has the WLP in degree 3.

The ideal fails to have the WLP in degree 5.

Geometric reasoning explains the WLP behavior.

Abstract

Here we explain geometrically why the ideal I = (L 3 1 ,. .. , L 3 8) $\subset$ C[x 0 ,. .. , x 6 ] has the WLP in degree 3 and why it fails to have it in degree 5.