On 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 $H_{norm}(X;D)=\frac{\hat{h}(X)}{(n+1)!}{D}^{n+1}+o(D^{n+1})$ where $\hat{h}(X)$ is the normalized height of $X$. This gives a positive answer to a question raised by Philippon and Sombra.

arXiv commentAccepted for publication in Algebra and Number Theory Journal