Point-curve incidences in the complex plane

TL;DR

This paper extends incidence bounds to complex algebraic curves, combining algebraic and differential geometry techniques to establish new bounds on point-curve incidences in the complex plane.

Contribution

It proves a nearly optimal incidence bound for points and algebraic curves in the complex plane, adapting real-plane techniques to complex geometry.

Findings

Established an incidence bound of $O_\varepsilon(m^{rac{k}{2k-1}+\varepsilon}n^{rac{2k-2}{2k-1}}+m+n)$ for complex algebraic curves.

Developed a key lemma controlling complex curves within real hypersurfaces, useful for incidence theorems over $\\mathbb{C}$.

Combined algebraic and differential geometry tools to achieve the main result.

Abstract

We prove an incidence theorem for points and curves in the complex plane. Given a set of $m$ points in ${\mathbb R}^2$ and a set of $n$ curves with $k$ degrees of freedom, Pach and Sharir proved that the number of point-curve incidences is $O\big(m^{\frac{k}{2k-1}}n^{\frac{2k-2}{2k-1}}+m+n\big)$. We establish the slightly weaker bound $O_\varepsilon\big(m^{\frac{k}{2k-1}+\varepsilon}n^{\frac{2k-2}{2k-1}}+m+n\big)$ on the number of incidences between $m$ points and $n$ (complex) algebraic curves in ${\mathbb C}^2$ with $k$ degrees of freedom. We combine tools from algebraic geometry and differential geometry to prove a key technical lemma that controls the number of complex curves that can be contained inside a real hypersurface. This lemma may be of independent interest to other researchers proving incidence theorems over ${\mathbb C}$.