American Journal of Mathematics

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 ${\cal L}$ be a collection of $n$ space curves in $k^3$, with $n\ll ({\rm char}(k))^2$ or ${\rm char}(k)=0$. 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 ${\cal L}$ 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.