On the normalized arithmetic Hilbert function

TL;DR

This paper proves that the normalized arithmetic Hilbert function of a subvariety in projective space asymptotically behaves like a polynomial involving the normalized height, confirming a conjecture by Philippon and Sombra.

Contribution

It establishes the asymptotic expansion of the normalized arithmetic Hilbert function in terms of the normalized height, answering a question posed by Philippon and Sombra.

Findings

The normalized arithmetic Hilbert function has a specific asymptotic expansion.

The leading term of the expansion involves the normalized height of the variety.

This confirms a conjecture about the behavior of the normalized arithmetic Hilbert function.

Abstract

Let $X$ be a subvariety of dimension n of the projective space over $\overline{\mathbb{Q}}$, and $H_{norm}(X;D)$ the normalized arithmetic Hilbert function of $X$ introduced by Philippon and Sombra. We show that this function admits the following asymptotic expansion $H_{norm}(X;D) = \frac{\widehat{h}(X)}{(n + 1)!}D^{n+1} + o(D^{n+1})$ where $\widehat{h}(X)$ is the normalized height of $X$. This gives a positive answer to a question raised by Philippon and Sombra.