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 $\lesssim_n |\mathfrak{L}|^{\frac{n}{n-1}}$ joints formed by $\mathfrak{L}$. We say that a point $x \in \mathbb{F}^n$ is a multijoint formed by the finite collections $\mathfrak{L}_1,\ldots,\mathfrak{L}_n$ 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 $n \geq 3$, there are $\lesssim_n \frac{|\mathfrak{L}|^{\frac{n}{n-1}}}{k^{\frac{n+1}{n-1}}}+\frac{|\mathfrak{L}|}{k}$ generic joints formed by $\mathfrak{L}$, each lying in $\sim k$ lines of $\mathfrak{L}$. 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.