On the ampleness and bigness of non-integral divisors

TL;DR

This paper investigates how classical notions of ampleness and bigness for divisors can be extended to non-integral divisors by examining associated line bundles and their properties.

Contribution

It provides a framework for extending classical divisor positivity criteria to non-integral divisors using line bundle constructions.

Findings

Characterizations of ampleness extended to non-integral divisors

Criteria for bigness adapted to non-integral divisors

Connections between divisor properties and associated line bundles

Abstract

Given a Weil non-integral divisor $D$, it is natural to associate it the line bundle of its integral part $\mathcal{O}_X([D])$. In this work we study which of the classical characterizations of ample and big divisors can be extended to non-integral divisors via the corresponding line bundles.