Paper detail

Incidences between points and curves with almost two degrees of freedom

We study incidences between points and algebraic curves in three dimensions, taken from a family $C$ of curves that have almost two degrees of freedom, meaning that every pair of curves intersect in $O(1)$ points, for any pair of points $p$, $q$, there are only $O(1)$ curves of $C$ that pass through both points, and a pair $p$, $q$ of points admit a curve of $C$ that passes through both of them iff $F(p,q)=0$ for some polynomial $F$. We study two specific instances, one involving unit circles in $R^3$ that pass through some fixed point (so called anchored unit circles), and the other involving tangencies between directed points (points and directions) and circles in the plane; a directed point is tangent to a circle if the point lies on the circle and the direction is the tangent direction. A lifting transformation of Ellenberg et al. maps these tangencies to incidences between points and curves in three dimensions. In both instances the curves in $R^3$ have almost two degrees of freedom. We show that the number of incidences between $m$ points and $n$ anchored unit circles in $R^3$, as well as the number of tangencies between $m$ directed points and $n$ arbitrary circles in the plane, is $O(m^{3/5}n^{3/5}+m+n)$. We derive a similar incidence bound, with a few additional terms, for more general families of curves in $R^3$ with almost two degrees of freedom. The proofs follow standard techniques, based on polynomial partitioning, but face a novel issue involving surfaces that are infinitely ruled by the respective family of curves, as well as surfaces in a dual 3D space that are infinitely ruled by the respective family of suitably defined dual curves. The general bound that we obtain is $O(m^{3/5}n^{3/5}+m+n)$ plus additional terms that depend on how many curves or dual curves can lie on an infinitely-ruled surface.