论文标题
立方超曲面中的线性子空间
Linear subspaces in cubic hypersurfaces
论文作者
论文摘要
我们证明，对于任何立方多项式级别等级$ r $，在相应的超表面中包含的所有最小编成imimension的线性子空间的相交具有码头$ \ le r^2+\ frac {（r+1）^2}^2}^2} {4} {4} {4} {4} {4}+r $。这是从独立利益的以下结果中得出的。考虑$ k [x_1，\ ldots，x_n] $的线性理想的交叉点$ i $，带有$ \ dim p_i \ le r $。然后，$ i $的二次发电机的数量为$ \ le r^2 $。
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 $\le r^2$.
