Incidence bounds on multijoints and generic joints

A point $x\in {\mathbb{F}}^n$ is a joint formed by a finite collection $\mathfrak{L}$ of lines in ${\mathbb{F}}^n$ if there exist at least $n$ lines in $\mathfrak{L}$ through $x$ that span ${\mathbb{F}}^n$. It is known that there are ${\lesssim }_n\mathrm{\mid }\mathfrak{L}{\mathrm{\mid }}^{\frac{n}{n-1}}$ joints formed by $\mathfrak{L}$. We say that a point $x\in {\mathbb{F}}^n$ is a multijoint formed by the finite collections ${\mathfrak{L}}_1,\dots ,{\mathfrak{L}}_n$ of lines in ${\mathbb{F}}^n$ if there exist at least $n$ lines through $x$, one from each collection, spanning ${\mathbb{F}}^n$. We show that there are $\lesssim_n (|\mathfrak{L}_1|\cdots |\mathfrak{L}_n|)^{\frac{1}{n-1}}$ such points for any field $\mathbb{F}$ and $n=3$, as well as for $\mathbb{F}=\mathbb{R}$ and any $n \geq 3$. Moreover, we say that a point $x\in {\mathbb{F}}^n$ is a generic joint formed by a finite collection $\mathfrak{L}$ of lines in ${\mathbb{F}}^n$ if each $n$ lines of $\mathfrak{L}$ through $x$ form a joint there. We show that, for $\mathbb{F}=\mathbb{R}$ and any $n\ge 3$, there are $\lesssim_n \frac{|\mathfrak{L}|^{\frac{n}{n-1}}}{k^{\frac{n+1}{n-1}}}+\frac{|\mathfrak{L}|}{k}$genericjointsformedby$\mathfrak{L}$,eachlyingin$\sim k$linesof$\mathfrak{L}$. This result generalises, to all dimensions, a (very small) part of the main point-line incidence theorem in ${\mathbb{R}}^3$ in \cite{Guth_Katz_2010} by Guth and Katz. Finally, we generalise our results in ${\mathbb{R}}^n$ to the case of multijoints and generic joints formed by real algebraic curves.