On weighted bounded negativity for rational surfaces

The weighted bounded negativity conjecture considers a smooth projective surface X and looks for a common lower bound on the quotients C²/(D· C)², where C runs over the integral curves on X and D over the big and nef divisors on X such that D · C >0. We focus our study on rational surfaces Z. Setting π: Z → Z₀ a composition of blowups giving rise to Z, where Z₀ is the projective plane or a Hirzebruch surface, we give a common lower bound on C²/(H^* · C)² whenever H^* is the pull-back of a nef divisor H on Z₀. In addition, we prove that, only in the case when a nef divisor D on Z approaches the boundary of the nef cone, the quotients C²/(D· C)² could tend to minus infinity.