On weighted bounded negativity for rational surfaces

TL;DR

This paper investigates a conjecture about bounded negativity on rational surfaces, establishing lower bounds for certain curve divisors and analyzing conditions under which these bounds can tend to negative infinity.

Contribution

It provides a lower bound for the quotients involving curves and nef divisors on rational surfaces and characterizes when these quotients can become arbitrarily negative.

Findings

Established a lower bound for C^2/(H^* · C)^2 on rational surfaces.

Showed that quotients can tend to minus infinity only when nef divisors approach the boundary of the nef cone.

Focused on surfaces obtained by blowups of the projective plane or Hirzebruch surfaces.

Abstract

The weighted bounded negativity conjecture considers a smooth projective surface $X$ and looks for a common lower bound on the quotients $C^2/(D\cdot C)^2$, 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 $\pi: Z \rightarrow Z_0$ 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 $C^2/(H^* \cdot C)^2$ whenever $H^*$ 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 $C^2/(D\cdot C)^2$ could tend to minus infinity.