Algebraic curves, rich points, and doubly-ruled surfaces

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 $n<\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}<(\mathrm{char}(k){)}^2$ or $\mathrm{char}(k)=0$. Then either A) there are at most $O(n^{3\mathrm{/}2})$ points in $k^3$ hit by at least two curves, or B) at least $\mathrm{\Omega }(n^{1\mathrm{/}2})$ 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.