An arithmetic Hilbert–Samuel theorem for pointed stable curves

Freixas i Montplet, Gerard. "An arithmetic Hilbert–Samuel theorem for pointed stable curves." Journal of the European Mathematical Society 014.2 (2012): 321-351. <http://eudml.org/doc/277284>.

@article{FreixasiMontplet2012,
abstract = {Let $(\mathcal \{O\},\sum , F_\infty )$ be an arithmetic ring of Krull dimension at most $1,\ S=\text\{Spec\}(\mathcal \{O\})$ and $(\mathcal \{X\}\rightarrow S; \sigma _1,\ldots , \sigma _n)$ a pointed stable curve. Write $\mathcal \{U\}=\mathcal \{X\} \setminus \cup _j \sigma _j(S)$. For every integer $k>0$, the invertible sheaf $\omega ^\{k+1\}_\{\mathcal \{X\}/S\} (k\sigma _1+\ldots + k\sigma _n)$ inherits a singular hermitian structure from the hyperbolic metric on the Riemann surface $\mathcal \{U\}_\infty $. In this article we define a Quillen type metric $\left\Vert \cdot \right\Vert _Q$ on the determinant line $\lambda _\{k+1\}=\lambda \omega ^\{k+1\}_\{\mathcal \{X\}/S\}$$(k\sigma _1 + \ldots + k\sigma _n)$ and compute the arithmetic degree of $(\lambda _\{k+1\}, \left\Vert \cdot \right\Vert _Q)$ 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 $(\lambda _\{k+1\}, \left\Vert \cdot \right\Vert _\{L^2\})$) admits an asymptotic expansion in $k$, whose leading coefficient is given by the arithmetic self-intersection of $(\omega _\{\mathcal \{X\}/S\}(k\sigma _1 + \ldots + k\sigma _n), \left\Vert \cdot \right\Vert _\{hyp\})$. Here $\left\Vert \cdot \right\Vert _\{L^2\}$ and $\left\Vert \cdot \right\Vert _\{hyp\}$ 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.},
author = {Freixas i Montplet, Gerard},
journal = {Journal of the European Mathematical Society},
keywords = {Arakelov theory; pointed stable curve; Mumford isomorphism; hyperbolic metric; Quillen metric; Selberg zeta function; Arakelov theory; pointed stable curve; Mumford isomorphism; hyperbolic metric; Quillen metric; Selberg zeta function},
language = {eng},
number = {2},
pages = {321-351},
publisher = {European Mathematical Society Publishing House},
title = {An arithmetic Hilbert–Samuel theorem for pointed stable curves},
url = {http://eudml.org/doc/277284},
volume = {014},
year = {2012},
}

TY - JOUR
AU - Freixas i Montplet, Gerard
TI - An arithmetic Hilbert–Samuel theorem for pointed stable curves
JO - Journal of the European Mathematical Society
PY - 2012
PB - European Mathematical Society Publishing House
VL - 014
IS - 2
SP - 321
EP - 351
AB - Let $(\mathcal {O},\sum , F_\infty )$ be an arithmetic ring of Krull dimension at most $1,\ S=\text{Spec}(\mathcal {O})$ and $(\mathcal {X}\rightarrow S; \sigma _1,\ldots , \sigma _n)$ a pointed stable curve. Write $\mathcal {U}=\mathcal {X} \setminus \cup _j \sigma _j(S)$. For every integer $k>0$, the invertible sheaf $\omega ^{k+1}_{\mathcal {X}/S} (k\sigma _1+\ldots + k\sigma _n)$ inherits a singular hermitian structure from the hyperbolic metric on the Riemann surface $\mathcal {U}_\infty $. In this article we define a Quillen type metric $\left\Vert \cdot \right\Vert _Q$ on the determinant line $\lambda _{k+1}=\lambda \omega ^{k+1}_{\mathcal {X}/S}$$(k\sigma _1 + \ldots + k\sigma _n)$ and compute the arithmetic degree of $(\lambda _{k+1}, \left\Vert \cdot \right\Vert _Q)$ 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 $(\lambda _{k+1}, \left\Vert \cdot \right\Vert _{L^2})$) admits an asymptotic expansion in $k$, whose leading coefficient is given by the arithmetic self-intersection of $(\omega _{\mathcal {X}/S}(k\sigma _1 + \ldots + k\sigma _n), \left\Vert \cdot \right\Vert _{hyp})$. Here $\left\Vert \cdot \right\Vert _{L^2}$ and $\left\Vert \cdot \right\Vert _{hyp}$ 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.
LA - eng
KW - Arakelov theory; pointed stable curve; Mumford isomorphism; hyperbolic metric; Quillen metric; Selberg zeta function; Arakelov theory; pointed stable curve; Mumford isomorphism; hyperbolic metric; Quillen metric; Selberg zeta function
UR - http://eudml.org/doc/277284
ER -