An arithmetic Hilbert–Samuel theorem for pointed stable curves

Gerard Freixas i Montplet

Journal of the European Mathematical Society, Tome 14 (2012) no. 2, pp. 321-351.

Résumé

Let (O,∑,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 ωX/Sk+1​(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(ωX/Sk+1​(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 L2 metric and the dual of the hyperbolic metric, respectively. Examples of application are given for pointed stable curves of genus 0.

Détail

DOI : 10.4171/jems/304

Classification : 14-XX, 32-XX, 00-XX
Keywords: Arakelov theory, pointed stable curve, Mumford isomorphism, hyperbolic metric, Quillen metric, Selberg zeta function