Incidence bounds on multijoints and generic joints

Abstract

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 ${\mathfrak{L}}_1,\dots ,{\mathfrak{L}}_n$ 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 $n\ge 3$. 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 ${\lesssim }_n\frac{|\mathfrak{L}{|}^{\frac{n}{n-1}}}{k^{\frac{n+1}{n-1}}}+\frac{|\mathfrak{L}|}{k}$ 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.

Some errors corrected. Theorem 4.4 is now slightly stronger than its previous version. To appear in Discrete Comput. Geom

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• Joints of varieties