On the joints problem with multiplicities

TL;DR

This paper applies Kollár's theorem on the arithmetic genus of curves to establish bounds on the number of joints with multiplicities, confirming a conjecture of Carbery in generic cases.

Contribution

It introduces a novel application of Kollár's theorem to the joints problem, providing bounds in the context of multiplicities and resolving a conjecture in the generic case.

Findings

Bound on the number of joints with multiplicities

Affirmative answer to Carbery's conjecture in the generic case

Application of algebraic geometry to combinatorial geometry

Abstract

In this short note we apply a recent theorem of Koll\'ar about the arithmetic genus of curves to give a bound on the number of joints weighted by the multiplicities. This gives an affirmative answer to a conjecture of Carbery in the generic case.