An arithmetic Hilbert–Samuel theorem for pointed stable curves

Abstract

Let $(\mathcal{O},\sum ,F_{\infty })$ be an arithmetic ring of Krull dimension at most 1, ${\mathcal{S}}_{=}\mathrm{Spec}\mathcal{O}$ and $(\mathcal{X}\to \mathcal{S};{\sigma }_1,...,{\sigma }_n)$ a pointed stable curve. Write $\mathcal{U}=\mathcal{X}\backslash {\bigcup }_j{\sigma }_j(\mathcal{S})$. For every integer $k\ge 0$, the invertible sheaf ${\omega }_{X/\mathcal{S}}^{k+1}(k{\sigma }_1+...+k{\sigma }_n)$ inherits a singular hermitian structure from the hyperbolic metric on the Riemann surface ${\mathcal{U}}_{\infty }$. In this article we define a Quillen type metric $\Vert \bullet {\Vert }_Q$ on the determinant line ${\lambda }_{k+1}=\lambda ({\omega }_{X/\mathcal{S}}^{k+1}(k{\sigma }_1+...+k{\sigma }_n))$ and compute the arithmetic degree of $({\lambda }_{k+1},\Vert \bullet {\Vert }_Q)$ 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 $({\lambda }_{k+1},\Vert \bullet {\Vert }_{L^2})$ admits an asymptotic expansion in $k$, whose leading coefficient is given by the arithmetic self-intersection of $({\omega }_{X/\mathcal{S}}({\sigma }_1+...+{\sigma }_n),\Vert \bullet {\Vert }_{\mathrm{hyp}})$. Here $\Vert \bullet {\Vert }_{L^2}$ and $\Vert \bullet {\Vert }_{\mathrm{hyp}}$ 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.