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Algebraic curves, rich points, and doubly-ruled surfaces

Author(s)
Guth, Lawrence; Zahl, Joshua
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Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/
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Abstract
We study the structure of collections of algebraic curves in three dimensions that have many curve-curve incidences. In particular, let k beafieldandlet L be a collection of n space curves in k3, with n ≪ (char(k))2 or char(k) =0. Then either (a) there are at most O(n3/2) points in k3 hit by at least two curves, or (b) at least Ω(n1/2 ) curves from L must lie on a bounded-degree surface, and many of the curves must form two “rulings” of this surface. We also develop several new tools including a generalization of the classical flecnode polynomial of Salmon and new algebraic techniques for dealing with this generalized flecnode polynomial.
Date issued
2018-10
URI
https://hdl.handle.net/1721.1/122976
Department
Massachusetts Institute of Technology. Department of Mathematics
Journal
American Journal of Mathematics
Publisher
Project Muse
Citation
Guth, Larry and Joshua Zahl. "Algebraic curves, rich points, and doubly-ruled surfaces." American Journal of Mathematics 140, 5 (October 2018): 1187-1229 ©2018 Project MUSE
Version: Original manuscript
ISSN
1080-6377

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