An arithmetic Hilbert–Samuel theorem for pointed stable curves

  • Gerard Freixas i Montplet

    Institut de Mathématiques de Jussieu, Paris, France


Let be an arithmetic ring of Krull dimension at most 1, and a pointed stable curve. Write . For every integer , the invertible sheaf inherits a singular hermitian structure from the hyperbolic metric on the Riemann surface . In this article we define a Quillen type metric on the determinant line and compute the arithmetic degree of 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 admits an asymptotic expansion in , whose leading coefficient is given by the arithmetic self-intersection of . Here and denote the 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