Geodesic
An arithmetic Hilbert–Samuel theorem for pointed stable curves
Journal of the European Mathematical Society, Tome 14 (2012) no. 2, pp. 321-351

Voir la notice de l'article provenant de la source EMS Press

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.
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 
@article{JEMS_2012_14_2_a0,
     author = {Gerard Freixas i Montplet},
     title = {An arithmetic {Hilbert{\textendash}Samuel} theorem for pointed stable curves},
     journal = {Journal of the European Mathematical Society},
     pages = {321--351},
     publisher = {mathdoc},
     volume = {14},
     number = {2},
     year = {2012},
     doi = {10.4171/jems/304},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/304/}
}
TY  - JOUR
AU  - Gerard Freixas i Montplet
TI  - An arithmetic Hilbert–Samuel theorem for pointed stable curves
JO  - Journal of the European Mathematical Society
PY  - 2012
SP  - 321
EP  - 351
VL  - 14
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.4171/jems/304/
DO  - 10.4171/jems/304
ID  - JEMS_2012_14_2_a0
ER  - 
%0 Journal Article
%A Gerard Freixas i Montplet
%T An arithmetic Hilbert–Samuel theorem for pointed stable curves
%J Journal of the European Mathematical Society
%D 2012
%P 321-351
%V 14
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.4171/jems/304/
%R 10.4171/jems/304
%F JEMS_2012_14_2_a0
Gerard Freixas i Montplet. An arithmetic Hilbert–Samuel theorem for pointed stable curves. Journal of the European Mathematical Society, Tome 14 (2012) no. 2, pp. 321-351. doi: 10.4171/jems/304

Cité par Sources :