Harbourne constants, pull-back clusters and ramified morphisms

TL;DR

This paper investigates how ramified morphisms influence Harbourne constants of divisors, introduces a pullback operation for clusters of points, and constructs curve configurations with new extremal Harbourne index values.

Contribution

It introduces the pullback of weighted clusters under morphisms and applies this to find curve configurations with Harbourne indices surpassing previous bounds.

Findings

Constructed curve configurations with Harbourne index close to -25/7

Described properties of pullback of clusters under morphisms

Analyzed the impact of ramified morphisms on Harbourne constants

Abstract

We describe the effect of ramified morphisms on Harbourne constants of reduced effective divisors. With this goal, we introduce the pullback of a weighted cluster of infinitely near points under a dominant morphism between surfaces, and describe some of its basic properties. As an application, we describe configurations of curves with transversal intersections and $H$-index arbitarily close to $-25/7\simeq -3.571$, smaller than any previously known result.