  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 $\le r^2$.

Algebraic Geometry;Commutative Algebra