Bounded negativity and Harbourne constants on ruled surfaces

Abstract

Let X be a smooth projective surface and let $${\mathcal {C}}$$ be an arrangement of curves on X. The Harbourne constant of $${\mathcal {C}}$$ was defined as a way to investigate the occurrence of curves of negative self-intersection on blow ups of X. This is related to the bounded negativity conjecture which predicts that the self-intersection number of all reduced curves on a surface is bounded below by a constant. We consider a geometrically ruled surface X over a smooth curve and give lower bounds for the Harbourne constants of transversal arrangements of curves on X. We also define a global Harbourne constant as the infimum of Harbourne constants for arrangements of a specific type and give a lower bound for it.