Linear subspaces in cubic hypersurfaces

Alexander Polishchuk; Chen Wang

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arXiv · 2206.09121

Linear subspaces in cubic hypersurfaces

Alexander Polishchuk·Chen Wang

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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 $\le r^2$.

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Alexander Polishchuk, Chen Wang. 2022-06-18. Linear subspaces in cubic hypersurfaces. https://arxiv.org/abs/2206.09121

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