Joints of Varieties
Authors
Tidor, J
Yu, HHH
Zhao, Y
Publication Date
2022-04Journal Title
Geometric and Functional Analysis
ISSN
1016-443X
Publisher
Springer Science and Business Media LLC
Volume
32
Issue
2
Pages
302-339
Language
en
Type
Article
This Version
VoR
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Tidor, J., Yu, H., & Zhao, Y. (2022). Joints of Varieties. Geometric and Functional Analysis, 32 (2), 302-339. https://doi.org/10.1007/s00039-022-00597-5
Description
Funder: Massachusetts Institute of Technology (MIT)
Abstract
<jats:title>Abstract</jats:title><jats:p>We generalize the Guth–Katz joints theorem from lines to varieties. A special case says that <jats:italic>N</jats:italic> planes (2-flats) in 6 dimensions (over any field) have <jats:inline-formula><jats:alternatives><jats:tex-math>$$O(N^{3/2})$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>O</mml:mi>
<mml:mo>(</mml:mo>
<mml:msup>
<mml:mi>N</mml:mi>
<mml:mrow>
<mml:mn>3</mml:mn>
<mml:mo>/</mml:mo>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula> joints, where a joint is a point contained in a triple of these planes not all lying in some hyperplane. More generally, we prove the same bound when the set of <jats:italic>N</jats:italic> planes is replaced by a set of 2-dimensional algebraic varieties of total degree <jats:italic>N</jats:italic>, and a joint is a point that is regular for three varieties whose tangent planes at that point are not all contained in some hyperplane. Our most general result gives upper bounds, tight up to constant factors, for joints with multiplicities for several sets of varieties of arbitrary dimensions (known as Carbery’s conjecture). Our main innovation is a new way to extend the polynomial method to higher dimensional objects, relating the degree of a polynomial and its orders of vanishing on a given set of points on a variety.</jats:p>
Keywords
Article
Identifiers
s00039-022-00597-5, 597
External DOI: https://doi.org/10.1007/s00039-022-00597-5
This record's URL: https://www.repository.cam.ac.uk/handle/1810/336148
Rights
Licence:
http://creativecommons.org/licenses/by/4.0/
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