An arithmetic Hilbert–Samuel theorem for pointed stable curves

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 ‖ ∙ ‖Q on the determinant line λ__k+1 = λ__ω__k+1_Χ/S (_k__σ_1 + ... + 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.