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.


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


Zbl 0773.11047
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