## Heights of zero-dimensional subschemes of projective space. (Hauteurs des sous-schémas de dimension nulle.)(French)Zbl 1076.14030

Let $$K$$ be a number field, $$\mathcal O_K$$ its ring of integers, $$\mathcal X$$ a projective scheme over $$\text{Spec}\mathcal O_K$$ with smooth generic fiber, $$\overline{\mathcal L}$$ an ample invertible sheaf on $$\mathcal X$$ with positively-curved hermitian metric, and $$\mu$$ a strictly positive measure on $$\mathcal X(\mathbb C)$$ invariant under complex conjugation. The $$L^2(\mu)$$ metrics on $$\Gamma(\mathcal X,\mathcal L^{\otimes n})$$ provide hermitian metrics on these $$\mathcal O_K$$-modules for all $$n\in\mathbb N$$.
Let $$\Sigma$$ be a closed subscheme of $$\mathcal X$$ of relative dimension $$0$$. The image of the restriction map $$\Gamma(\mathcal X,\mathcal L^{\otimes n})\to \Gamma(\Sigma,\mathcal L^{\otimes n}\bigr| _\Sigma)$$ is then given the quotient hermitian metrics, and its resulting Arakelov degree defines the quantity $$h(\Sigma;n)$$. The latter may be regarded as an arithmetic Hilbert-Samuel function. This paper first proves an explicit asymptotic formula for $$h(\Sigma;n)$$ as $$n\to\infty$$, in the case where $$\mathcal X$$ is the projective space associated to a hermitian vector sheaf over $$\text{Spec}\mathcal O_K$$ and $$\overline{\mathcal L}=\overline{\mathcal O(1)}$$.
If one further restricts to $$\mathcal X=\mathbb P^n_K$$ and $$\Sigma$$ reduced, associated to points $$p_1,\dots,p_l$$, then work of M. Laurent [in: Approximations diophantiennes et nombres transcendants, C. R. Colloq., Luminy/ Fr. 1990, 215–238 (1992; Zbl 0773.11047)] shows that if $$A_n$$ is the map $$K[X_0,\dots,X_N]_n\to K^l$$ given by evaluation at the $$p_i$$, then $$h\bigl(\bigwedge^l A_n\bigr)=n\sum_{i=1}^l h(p_i) +\chi(\mathcal O_\Sigma)+o(1)$$. The present paper shows that $$\chi(\mathcal O_\Sigma)$$ is given explicitly in terms of the difference between the scheme $$\Sigma$$ and its normalization.

### MSC:

 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 11G50 Heights

Zbl 0773.11047
Full Text:

### References:

 [1] Heights of projective varieties and positive Green forms, Journal Amer. Math. Soc, 7, 903-1027, (1994) · Zbl 0973.14013 [2] Grothendieck Duality and Base Change, (1750), Springer-Verlag · Zbl 0992.14001 [3] Intersection theory, Vol. 2, (1998), Springer-Verlag · Zbl 0885.14002 [4] Techniques de construction et théorèmes d’existence en géométrie algébrique. IV : LES schémas de Hilbert, Fondements de la géométrie algébrique, exposé 221, (1960) · Zbl 0236.14003 [5] Éléments de géométrie algébrique, 4, 8, 11, 17, 20, 24, 28, 32, (19601967) [6] Residues and Duality, 20, (1966), Springer-Verlag · Zbl 0212.26101 [7] Rational Curves on Algebraic Varieties, Vol. 32, (1996), Springer-Verlag · Zbl 0877.14012 [8] Hauteur de matrices d’interpolation, Approximations Diophantiennes et Nombres Transcendants (Luminy, 1990), 215-238, (1992), de Gruyter · Zbl 0773.11047 [9] Local Euler characteristics, Inventiones Math., 9, 201-234, (1970) · Zbl 0191.19202 [10] Lectures on Curves on an Algebraic Surface (with a section by G. Bergman), 59, (1966), Princeton University Press · Zbl 0187.42701 [11] On heights of algebraic subspaces and Diophantine approximations, Annals of Math, 85, 430-472, (1967) · Zbl 0152.03602 [12] Corps locaux, (1968), Hermann · Zbl 0137.02601
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