Vol. 9, No. 10, 2015

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On the normalized arithmetic Hilbert function

Mounir Hajli

Vol. 9 (2015), No. 10, 2293–2302
Abstract

Let X ¯N be a subvariety of dimension n, and let norm(X; ) be the normalized arithmetic Hilbert function of X introduced by Philippon and Sombra. We show that this function admits the asymptotic expansion

norm(X;D) = ĥ(X) (n + 1)!Dn+1 + o(Dn+1),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