Bounded negativity, Harbourne constants and transversal arrangements of curves

The Bounded Negativity Conjecture predicts that for every complex projective surface $X$ there exists a number $b(X)$ such that $C^2\ge -b(X)$ holds for all reduced curves $C\subset X$. For birational surfaces $f:Y\to X$ there have been introduced certain invariants (Harbourne constants) relating to the effect the numbers $b(X)$, $b(Y)$ and the complexity of the map $f$. These invariants have been studied previously when $f$ is the blowup of all singular points of an arrangement of lines in ${\mathbb{P}}^2$, of conics and of cubics. In the present note we extend these considerations to blowups of ${\mathbb{P}}^2$ at singular points of arrangements of curves of arbitrary degree $d$. We also considerably generalize and modify the approach witnessed so far and study transversal arrangements of sufficiently positive curves on arbitrary surfaces with the non-negative Kodaira dimension.