On the Bounded Negativity Conjecture and Singular Plane Curves

Abstract

We show that two questions raised by Brian Harbourne related to the bounded negativity conjecture and singular plane curves have a negative answer in some cases. For rational curves having only ordinary singularities, this question is shown to be related to strong new bounds on the number of singularities of multiplicity greater or equal to 3 such a curve may have. This fact suggests a conjecture on the non-existence of rational curves of degree d > 8 having only ordinary triple points as singularities. We also give lower bounds for theH-constant H(C) in terms of the maximal multiplicity of the singularities of C, or when C has only singularities of type As with 1 ≤ s ≤ 5 and D4.