On the boundedness of the denominators in the Zariski decomposition on surfaces

TL;DR

This paper investigates the relationship between bounded denominators in Zariski decompositions and the bounded negativity of curves on algebraic surfaces, establishing their equivalence and providing explicit bounds.

Contribution

It proves that bounded Zariski denominators are equivalent to the bounded negativity conjecture and offers explicit bounds linking the two concepts.

Findings

Bounded Zariski denominators are equivalent to bounded negativity of curves.

Explicit bounds are provided relating denominators and negativity.

The results connect geometric properties of surfaces with algebraic decomposition denominators.

Abstract

Zariski decompositions play an important role in the theory of algebraic surfaces. For making geometric use of the decomposition of a given divisor, one needs to pass to a multiple of the divisor in order to clear denominators. It is therefore an intriguing question whether the surface has a 'universal denominator' that can be used to simultaneously clear denominators in all Zariski decompositions on the surface. We prove in this paper that, somewhat surprisingly, this condition of bounded Zariski denominators is equivalent to the bounded negativity of curves that is addressed in the Bounded Negativity Conjecture. Furthermore, we provide explicit bounds for Zariski denominators and negativity of curves in terms of each other.