Effective bounds of linear series on algebraic varieties and arithmetic varieties

In this paper, we prove effective upper bounds for effective sections of line bundles on projective varieties and hermitian line bundles on arithmetic varieties in terms of the volumes. They are effective versions of the Hilbert-Samuel formula and the arithmetic Hilbert-Samuel formula. The treatments are high-dimensional generalizations of [25] and [26]. Similar results are obtained independently by Huayi Chen [7] with less explicit error terms.