Bounded negativity, Harbourne constants and transversal arrangements of curves

Abstract

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 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. The main result in this direction is stated in Theorem B. We also considerably generalize and modify the approach witnessed so far and study transversal arrangements of sufficiently positive curves on arbitrary surfaces with the nonnegative Kodaira dimension. The main result obtained in this general setting is presented in Theorem A.