An arithmetic Hilbert–Samuel theorem for pointed stable curves

Let (O, 6, F∞) be an arithmetic ring of Krull dimension at most 1, S = SpecO and (X → S; σ1, . . . , σn) a pointed stable curve. Write U = X \ ⋃ j σj (S). For every integer k ≥ 0, the invertible sheaf ωk+1 X /S (kσ1+ · · · + kσn) 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 λk+1 = λ(ω k+1 X /S (kσ1+· · ·+kσn)) and compute the arithmetic degree of (λk+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 (λk+1, ‖ · ‖L2) admits an asymptotic expansion in k, whose leading coefficient is given by the arithmetic self-intersection of (ωX /S (σ1 + · · · + σn), ‖ · ‖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.