Effective bound of linear series on arithmetic surfaces

TL;DR

This paper establishes an explicit upper bound on the number of sections of hermitian line bundles on arithmetic surfaces, leading to effective bounds on Faltings height and canonical bundle self-intersection.

Contribution

It provides the first effective version of the arithmetic Hilbert--Samuel formula for nef line bundles on arithmetic surfaces.

Findings

Effective upper bound on sections of hermitian line bundles

Lower bounds on Faltings height in terms of singular points

Bounds on self-intersection of the canonical bundle

Abstract

We prove an effective upper bound on the number of effective sections of a hermitian line bundle over an arithmetic surface. It is an effective version of the arithmetic Hilbert--Samuel formula in the nef case. As a consequence, we obtain effective lower bounds on the Faltings height and on the self-intersection of the canonical bundle in terms of the number of singular points on fibers of the arithmetic surface.