Linear subspaces of minimal codimension in hypersurfaces

2021/07/16 by David Kazhdan, Kazhdan, David, Alexander Polishchuk +1
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2107.08080

Abstract

Let k be a perfect field and let X⊂ \mathbb P^N be a hypersurface of degree d defined over k and containing a linear subspace L defined over an algebraic closure k with codim_\mathbb P^NL=r. We show that X contains a linear subspace L₀ defined over k with codim_\mathbb P^NL≤ dr. We conjecture that the intersection of all linear subspaces (over k) of minimal codimension r contained in X, has codimension bounded above only in terms of r and d. We prove this when either d≤ 3 or r≤ 2.

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