Curves in R4 and Two-Rich Points

We obtain a new bound on the number of two-rich points spanned by an arrangement of low degree algebraic curves in R. Specifically, we show that an arrangement of n algebraic curves determines at most Cǫn 4/3+3ǫ two-rich points, provided at most n curves lie in any low degree hypersurface and at most n curves lie in any low degree surface. This result follows from a structure theorem about arrangements of curves that determine many two-rich points.