What is the height of two points in the plane?

TL;DR

This paper studies the distribution of rational points on the Hilbert scheme of two points in the projective plane, providing explicit height functions and asymptotic formulas for counting rational points of bounded height.

Contribution

It introduces a two-parameter family of height functions and establishes asymptotic formulas for rational point counts within certain parameter ranges.

Findings

Explicit description of height functions $H_{s,t}$

Asymptotic formulas for rational point counts

Upper bounds for certain parameter ranges

Abstract

Here we describe the distribution of rational points on the Hilbert scheme of two points in the projective plane. More specifically, we explicitly describe a two-parameter family of height functions $H_{s, t}$, such that the height function associated to any projective embedding is equivalent to some $H_{s, t}$, up to multiplication by a bounded function. For a certain range of the parameters $(s, t)$, we prove an asymptotic formula for the number of rational points of bounded height, and for other $(s, t)$ we obtain an upper bound. The proof establishes an equivalence to a lattice point counting problem, which we solve using the geometry of numbers.