Linear subspaces of minimal codimension in hypersurfaces

David Kazhdan; Alexander Polishchuk

doi:10.4310/mrl.2023.v30.n1.a7

Linear subspaces of minimal codimension in hypersurfaces

David Kazhdan
, Alexander Polishchuk

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

Let k be a perfect field and let X ⊂ P^N be a hypersurface of degree d defined over k and containing a linear subspace L defined over k with codim_PNL = r. We show that X contains a linear subspace L₀ defined over k with codim_PNL ≤ 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.

Original language	English
Pages (from-to)	143-166
Number of pages	24
Journal	Mathematical Research Letters
Volume	30
Issue number	1
DOIs	https://doi.org/10.4310/mrl.2023.v30.n1.a7
State	Published - 2023

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© 2023 International Press of Boston, Inc.. All rights reserved.

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abstract = "Let k be a perfect field and let X ⊂ PN be a hypersurface of degree d defined over k and containing a linear subspace L defined over k with codimPNL = r. We show that X contains a linear subspace L0 defined over k with codimPNL ≤ 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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Linear subspaces of minimal codimension in hypersurfaces

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