Algebraic curves, rich points, and doubly-ruled surfaces

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 $n<<(\mathrm{char}(k){)}^2$ or $\mathrm{char}(k)=0$. 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.

34 pages, 0 figures. v2: Minor tweak to the definition of the affine Chow variety of curves. references updated and typos corrected

References in corpus (1)

• On the Erdos distinct distance problem in the plane

Cited by in corpus (6)

• Breaking the 3/2 barrier for unit distances in three dimensions
• Distinct distances in the complex plane
• Distinct and repeated distances on a surface and incidences between points and spheres
• Sphere tangencies, line incidences, and Lie's line-sphere correspondence
• Incidences with curves in R^d
• The geometry of the flex locus of a hypersurface