On the normalized arithmetic Hilbert function

Hajli, Mounir

doi:10.2140/ant.2015.9.2293

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On the normalized arithmetic Hilbert function

Mounir Hajli

Vol. 9 (2015), No. 10, 2293–2302

DOI: 10.2140/ant.2015.9.2293

Abstract

Let $X \subset ℙ_{‘ \bar{ℚ}}^{N}$ be a subvariety of dimension $n$ , and let $ℋ_{norm} (X; \cdot)$ be the normalized arithmetic Hilbert function of $X$ introduced by Philippon and Sombra. We show that this function admits the asymptotic expansion

ℋ_{norm} (X; D) = \frac{ĥ (X)}{(n + 1)!} D^{n + 1} + o (D^{n + 1}), \forall D ≫ 1,

where $ĥ (X)$ is the normalized height of $X$ . This gives a positive answer to a question raised by Philippon and Sombra.

Keywords

arithmetic Hilbert function, height

Mathematical Subject Classification 2010

Primary: 14G40

Secondary: 11G50, 11G35

Milestones

Received: 19 November 2014

Revised: 10 September 2015

Accepted: 15 October 2015

Published: 16 December 2015

Authors

Mounir Hajli
Institute of Mathematics, Academia Sinica 6F, Astronomy-Mathematics Building No. 1, Sec. 4, Roosevelt Road Taipei 10617 Taiwan

Vol. 9, No. 10, 2015