The Bounded Negativity Conjecture predicts that for every complex projective
surface X there exists a number b(X) such that C2≥−b(X) holds for
all reduced curves C⊂X. For birational surfaces f:Y→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 P2, of conics and of cubics. In the
present note we extend these considerations to blowups of P2 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.Comment: This is the final version, incorporating the suggestions of the
referee, to appear in Annales de l'Institut Fourier Grenobl