Curves in ${\mathbb{R}}^4$ 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 ${\mathbb{R}}^4$. Specifically, we show that an arrangement of $n$ algebraic curves determines at most $C_\epsilon n^{4/3+3\epsilon}$two-richpoints,providedatmost$n^{2/3+2\epsilon}$ curves lie in any low degree hypersurface and at most $n^{1\mathrm{/}3+ϵ}$ curves lie in any low degree surface. This result follows from a structure theorem about arrangements of curves that determine many two-rich points.