# Bounded volume denominators and bounded negativity

## TL;DR

This paper explores the relationship between bounded volume denominators and bounded negativity on smooth projective surfaces, establishing equivalences and implications related to divisor effectiveness and negativity bounds.

## Contribution

It proves that bounded volume denominators are equivalent to primitive bounded negativity and links this to the existence of negative classes requiring large multiples for effectivity.

## Key findings

- Bounded volume denominators are equivalent to primitive bounded negativity.
- Bounded negativity implies bounded volume denominators.
- Negative classes can require arbitrarily large multiples to become effective.

## Abstract

In this paper we study the question of whether on smooth projective surfaces the denominators in the volumes of big line bundles are bounded. In particular we investigate how this condition is related to bounded negativity (i.e., the boundedness of self-intersections of irreducible curves). Our first result shows that boundedness of volume denominators is equivalent to \emph{primitive bounded negativity}, which in turn is implied by bounded negativity. We connect this result to the study of semi-effective orders of divisors: Our second result shows that negative classes exist that become effective only after taking an arbitrarily large multiple.