# 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, $S_{=}SpecO$ and $(X→S;σ_{1},...,σ_{n})$ a pointed stable curve. Write $U=X\⋃_{j}σ_{j}(S)$. For every integer $k≥0$, the invertible sheaf $ω_{X/S}(kσ_{1}+...+kσ_{n})$ inherits a singular hermitian structure from the hyperbolic metric on the Riemann surface $U_{∞}$. In this article we define a Quillen type metric $‖∙‖_{Q}$ on the determinant line $λ_{k+1}=λ(ω_{X/S}(kσ_{1}+...+kσ_{n}))$ 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 $(λ_{k+1},‖∙‖_{L_{2}})$ admits an asymptotic expansion in $k$, whose leading coefficient is given by the arithmetic self-intersection of $(ω_{X/S}(σ_{1}+...+σ_{n}),‖∙‖_{hyp})$. Here $‖∙‖_{L_{2}}$ 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