Harbourne constants, pull-back clusters and ramified morphisms (Q2311956)

The paper is about the study of negative self-intersection for curves (i.e. reduced effective divisors) on an analytic surface \(S\); the case of plane curves is the most considered here. A ``folklore conjecture'' asserts that for any surface \(S\) here exists a \(b(S) \in \mathbb{N} \) such that for any curve \(C \subset S\) we have \(C^2 \geq -b(S)\). A related question is the following: Suppose that \(b(S) \in \mathbb{N} \) as in the conjecture exists and let \(S_\pi \rightarrow S\) be the composition of \(n\geq 1\) blow ups at a point; does it exist a number \(b(S,n) \in \mathbb{N} \) such that every \(C \subset S_\pi\) has \(C^2 \geq b(S,n)\)? Even for \(S = \mathbb{P}^2\) the answer is unknown \(\forall n\geq 9\). These sort of problems led to the definition of the Harbourne constant for a curve with only ordinary singularity \(C\subset \mathbb{P}^2\) of degree \(d\): \[ h(C) = \frac{d^2-\sum_{P\in\mathrm{Sing}(C)}\mu_P(C)^2}{|\mathrm{Sing}(C)|} \] where \(\mu_P(C)\) is the multiplicity of \(C\) at \(P\). The definition can be extended to any curve by considering also infinitely near points on \(S\). It is interesting to find curves with very negative \(h(C)\); the best known results are for curves with \(h(C)\) arbitrarly near to \(-225/67\): in this paper new examples are found with lower \(h(C)\) (arbitrarly near to \(-25/7\)). The main tool used in the paper is the study of finite morphisms \(f: S \rightarrow S'\) and how \(h(f^*(C))\) behaves with respect to \(h(C)\).