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arXiv · 1510.05493

On the normalized arithmetic Hilbert function

Mounir Hajli
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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.

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Mounir Hajli. 2015-10-19. On the normalized arithmetic Hilbert function. https://arxiv.org/abs/1510.05493

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