On weighted bounded negativity for rational surfaces

The weighted bounded negativity conjecture considers a smooth projective surface [Formula: see text] and looks for a common lower bound on the quotients [Formula: see text], where [Formula: see text] runs over the integral curves on [Formula: see text] and [Formula: see text] over the big and nef divisors on [Formula: see text] such that [Formula: see text]. We focus our study on rational surfaces [Formula: see text]. Setting [Formula: see text] a composition of blowups giving rise to [Formula: see text], where [Formula: see text] is the projective plane or a Hirzebruch surface, we give a common lower bound on [Formula: see text] whenever [Formula: see text] is the pull-back of a nef divisor [Formula: see text] on [Formula: see text]. In addition, we prove that, only in the case when a nef divisor [Formula: see text] on [Formula: see text] approaches the boundary of the nef cone, the quotients [Formula: see text] could tend to minus infinity.