Algebraic curves, rich points, and doubly-ruled surfaces

TL;DR

This paper investigates the structure of algebraic curves in three-dimensional space with many incidences, revealing that either incidences are limited or many curves lie on a special ruled surface, supported by new algebraic tools.

Contribution

It introduces a novel generalization of the flecnode polynomial and new algebraic techniques to analyze incidences among algebraic curves in 3D.

Findings

Either incidence points are bounded by O(n^{3/2})

Many curves must lie on a bounded-degree surface

Development of a generalized flecnode polynomial

Abstract

We study the structure of collections of algebraic curves in three dimensions that have many curve-curve incidences. In particular, let $k$ be a field and let $\mathcal{L}$ be a collection of $n$ space curves in $k^3$, with $n<\!\!<(\operatorname{char}(k))^2$ or $\operatorname{char}(k)=0$. Then either A) there are at most $O(n^{3/2})$ points in $k^3$ hit by at least two curves, or B) at least $\Omega(n^{1/2})$ curves from $\mathcal{L}$ 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.