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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.

On the Degree of Curves with Prescribed Multiplicities and Bounded Negativity