Algebraic curves, rich points, and doubly-ruled surfaces
Name
1503.02173.pdf
Description
Submitted version
Size
377.65 KB
Format
Adobe PDF
Checksum (MD5)
953407e434fe7659f2813dde7b66d5f5
Author(s)
Guth, Lawrence
Zahl, Joshua
Date Issued
October 2018
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
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1353/ajm.2018.0028
