Linear subspaces of minimal codimension in hypersurfaces
Abstract
Let $k$ be a perfect field and let $X\subset {\mathbb P}^N$ be a hypersurface of degree $d$ defined over $k$ and containing a linear subspace $L$ defined over an algebraic closure $\overline{k}$ with $\mathrm{codim}_{{\mathbb P}^N}L=r$. We show that $X$ contains a linear subspace $L_0$ defined over $k$ with $\mathrm{codim}_{{\mathbb P}^N}L\le dr$. We conjecture that the intersection of all linear subspaces (over $\overline{k}$) of minimal codimension $r$ contained in $X$, has codimension bounded above only in terms of $r$ and $d$. We prove this when either $d\le 3$ or $r\le 2$.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2021
- arXiv:
- arXiv:2107.08080
- Bibcode:
- 2021arXiv210708080K
- Keywords:
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- Mathematics - Algebraic Geometry;
- Mathematics - Commutative Algebra
- E-Print:
- 15 pages, v2 substantially rewritten: added Conjecture B and a result on hypersurfaces of rank 2