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arXiv · 2209.13030

What is the height of two points in the plane?

Jesse Leo Kass·Frank Thorne
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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.

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Jesse Leo Kass, Frank Thorne. 2022-09-26. What is the height of two points in the plane?. https://arxiv.org/abs/2209.13030

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