Linear subspaces in cubic hypersurfaces

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 \(\le r^{2}+{(r+1)^{2}\over{4}}+r\) $$\le r^2+\frac{(r+1{)}^2}{4}+r$$ 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[x₁, …, x_n], with dim P_i ≤ r. Then the number of quadratic generators of I is ≤ r².