Linear subspaces in cubic hypersurfaces

2022/06/18 by Alexander Polishchuk, Chen Wang, Polishchuk, Alexander +1
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2206.09121

openalex publication_date 2022/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ≤ r²+((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.

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