The numerical equivalence relation for height functions and ampleness and nefness criteria for divisors

TL;DR

This paper investigates the properties of Weil height functions related to divisors, introducing a fractional limit concept that encodes numerical information to determine ampleness, nefness, or pseudo-effectiveness of divisors.

Contribution

It defines a fractional limit of height functions that captures numerical divisor properties, providing criteria for ampleness, nefness, and pseudo-effectiveness.

Findings

Fractional limit $ ext{Flim}_D(E,U)$ encodes divisor properties.

Criteria for ampleness, nefness, and pseudo-effectiveness based on height functions.

New tool for numerical divisor classification.

Abstract

In this paper, we study properties of Weil height functions associated with numerically trivial divisors. It helps us to define the fractional limit of $h_E$ with respect to $h_D$ on $U$, with $D$ ample: \[ \Flim_D(E,U) := \liminf_{\substack{P \in U h_D(P) \rightarrow \infty}}\dfrac{h_E(P)}{h_D(P)}. \] The value of $\Flim_D(E,U)$ contains numerical information about a divisor $E$, enough to determine whether $E$ is ample, numerically effective or pseudo-effective.