×

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

Citations:

Zbl 0773.11047
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.