Harbourne constants, pull-back clusters and ramified morphisms

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 $-25\mathrm{/}7\simeq -3.571$, smaller than any previously known result.