On the normalized arithmetic Hilbert function

Abstract

Let X ⊂ P N Q be a subvariety of dimension n, and Hnorm(X; ·) the normalized arithmetic Hilbert function of X introduced by Philippon and Sombra. We show that this function admits the following asymptotic expansionwhere h(X) is the normalized height of X. This gives a positive answer to a question raised by Philippon and Sombra.

Explicit uniform estimate of arithmetic Hilbert-Samuel function of hypersurfaces

“…In [26], P. Philippon et M. Sombra proposed another definition of the arithmetic Hilbert-Samuel function, and they proved an asymptotic formula for the case of toric varieties (see [26,Théorème 0.1]). In [19], M. Hajli proved the same asymptotic formula for the case of general projective varieties with the definition in [26].…”