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 $C^2\mathrm{/}(D\cdot C{)}^2$, where $C$ runs over the integral curves on $X$ and $D$ over the big and nef divisors on $X$ such that $D\cdot C>0$. We focus our study on rational surfaces $Z$. Setting $\pi :Z\to Z_0$ a composition of blowups giving rise to $Z$, where $Z_0$ is the projective plane or a Hirzebruch surface, we give a common lower bound on $C^2\mathrm{/}(H^{\ast }\cdot C{)}^2$ whenever $H^{\ast }$ is the pull-back of a nef divisor $H$ on $Z_0$. 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 $C^2\mathrm{/}(D\cdot C{)}^2$ could tend to minus infinity.