Effective Bounds of Linear Series on Algebraic Varieties and Arithmetic Varieties

TL;DR

This paper establishes effective upper bounds for sections of line bundles on algebraic and arithmetic varieties, providing concrete versions of classical Hilbert--Samuel formulas to quantify the growth of sections.

Contribution

It introduces effective bounds for line bundle sections on algebraic and arithmetic varieties, extending classical formulas with explicit estimates.

Findings

Derived effective upper bounds in terms of volumes

Extended Hilbert--Samuel formula to effective versions

Applied results to algebraic and arithmetic varieties

Abstract

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.