Counting joints with multiplicities

Let $\mathfrak{L}$ be a collection of $L$ lines in ${\mathbb{R}}^3$ and $J$ the set of joints formed by $\mathfrak{L}$, i.e. the set of points each of which lies in at least 3 non-coplanar lines of $\mathfrak{L}$. It is known that $|J| \lesssim L^{3/2}$(firstprovedbyGuthandKatz)\mathrm{.}Foreachjoint$x \in J$, let the multiplicity $N(x)$ of $x$ be the number of triples of non-coplanar lines through $x$. We prove here that ${\sum }_{x\in J}N(x{)}^{1\mathrm{/}2}\lesssim L^{3\mathrm{/}2}$, while in the last section we extend this result to real algebraic curves of uniformly bounded degree in ${\mathbb{R}}^3$, as well as to curves in ${\mathbb{R}}^3$ parametrised by real polynomials of uniformly bounded degree.