  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.
