On the Degree of Curves with Prescribed Multiplicities and Bounded Negativity

Abstract

We provide a lower bound on the degree of curves of the projective plane $\mathbb {P}^2$ passing through the centers of a divisorial valuation $\nu $ of $\mathbb {P}^2$ with prescribed multiplicities, and an upper bound for the Seshadri-type constant of $\nu $, $\hat {\mu }(\nu )$, constant that is crucial in the Nagata-type valuative conjecture. We also give some results related to the bounded negativity conjecture concerning those rational surfaces having the projective plane as a relatively minimal model.