Linear subspaces in cubic hypersurfaces

TL;DR

This paper establishes an upper bound on the codimension of the intersection of all minimal codimension linear subspaces within cubic hypersurfaces, based on the polynomial's slice rank, and relates it to the number of quadratic generators in ideal intersections.

Contribution

It introduces a new bound linking the slice rank of cubic polynomials to the structure of their linear subspaces and ideal intersections, advancing understanding of cubic hypersurface geometry.

Findings

Bound on codimension of linear subspaces in cubic hypersurfaces based on slice rank

Limit on quadratic generators in intersections of linear ideals with bounded dimension

Connection between polynomial slice rank and geometric properties of hypersurfaces

Abstract

We prove that for any cubic polynomial of slice rank $r$, the intersection of all linear subspaces of minimal codimension contained in the corresponding hypersurface has codimension $\le r^2+\frac{(r+1)^2}{4}+r$ in the affine space. This is deduced from the following result of independent interest. Consider the intersection $I$ of linear ideals $(P_i)$ in $k[x_1,\ldots,x_n]$, with $\dim P_i\le r$. Then the number of quadratic generators of $I$ is $\le r^2$.