Effective bound of linear series on arithmetic surfaces

摘要

We prove effective upper bounds 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.