# An arithmetic Hilbert–Samuel theorem for pointed stable curves

### Gerard Freixas i Montplet

Institut de Mathématiques de Jussieu, Paris, France

## Abstract

Let (*O*,∑,*F_∞) be an arithmetic ring of Krull dimension at most 1, S = Spec O and (Χ → S; σ_1, ..., σ__n) a pointed stable curve. Write U = Χ \ ∪_j__σ__j(S). For every integer k > 0, the invertible sheaf ω__k+1_Χ/S*(_k__σ_1 + ... +

*k__σ__n*) inherits a singular hermitian structure from the hyperbolic metric on the Riemann surface

*U_∞. In this article we define a Quillen type metric ‖ ∙ ‖*(_k__σ_1 + ... +

*Q*on the determinant line*λ__k*+1 =*λ__ω__k*+1_Χ/S*k__σ__n*) and compute the arithmetic degree of (

*λ__k*+1, ‖ ∙ ‖

*Q*) by means of an analogue of the Riemann–Roch theorem in Arakelov geometry. As a byproduct, we obtain an arithmetic Hilbert–Samuel formula: the arithmetic degree of (

*λ__k*+1, ‖ ∙ ‖

*L2*)) admits an asymptotic expansion in

*k*, whose leading coefficient is given by the arithmetic self-intersection of (

*ω__Χ/S*(_k__σ_1 + ... +

*k__σ__n*), ‖ ∙ ‖

*hyp*). Here ‖ ∙ ‖

*L2*and ‖ ∙ ‖

*hyp*denote the _L_2 metric and the dual of the hyperbolic metric, respectively. Examples of application are given for pointed stable curves of genus 0.

## Cite this article

Gerard Freixas i Montplet, An arithmetic Hilbert–Samuel theorem for pointed stable curves. J. Eur. Math. Soc. 14 (2012), no. 2, pp. 321–351

DOI 10.4171/JEMS/304