Point-curve incidences in the complex plane

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 ${\mathbb C}$.Comment: The proof was significantly simplified, and now relies on the Picard-Lindelof theorem, rather than on foliation

The strong Massey vanishing conjecture for fields with virtual cohomological dimension at most 11

We show that a strong vanishing conjecture for $n$-fold Massey products holds for fields of virtual cohomological dimension at most $1$ using a theorem of Haran. We also prove the same for PpC fields, using results of Haran--Jarden.Comment: 10 pages. Originally this paper was the last section of our paper "The fibration method over real function fields". The referee asked us to remove it, but because of ArXiv policy we cannot post it as a separate paper. Revised version, with a new result suggested by the referee. Submitte

Arrangements of translates of a curve

We show that there are five types of planar curves such that arrangements of its translates are combinatorially equivalent to an arrangement of lines. These curves can be used to define norms giving constructions with many unit distances among points in the plane