Algebraic curves, rich points, and doubly-ruled surfaces
Author(s)Guth, Lawrence; Zahl, Joshua
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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.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
American Journal of Mathematics
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