On weighted bounded negativity for rational surfaces

TL;DR

This work advances the weighted bounded negativity conjecture (WBNC) for smooth projective rational surfaces by proving a computable global lower bound $\frac{C^2}{(H^*\cdot C)^2} \ge -A(Z)$ where $H^*$ is the pullback of a nef divisor on the base $Z_0$ and $A(Z)$ depends only on the configuration of blowups. It further shows that quotients can only diverge to $-\infty$ when divisors approach the nef cone boundary, and it provides a second, epsilon-dependent bound $\frac{C^2}{(D\cdot C)^2} \ge -\frac{A'(Z)}{\varepsilon^2}$ for divisors $D$ in neighborhoods $\Delta_{H^*}(Z,\varepsilon)$, unifying WBNC with a robust control mechanism. The results rely on the arrowed proximity graph (APG) and a polyhedral description of the effective cone for large $\delta$ in $\mathbb{F}_\delta$-type blowups, enabling effective computation of $A(Z)$. The paper also provides concrete examples showing how these bounds can be computed in practice, including configurations over $\mathbb{P}^2$ and various $\mathbb{F}_\delta$, and demonstrates that the bounds remain stable even as the Picard number grows. Overall, the methods yield explicit, computable criteria for bounded negativity on broad families of rational surfaces, significantly advancing the WBNC program in characteristic zero.

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 $π: Z \rightarrow Z_0$ 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.

Figures (3)

• Figure 2.1: Arrowed proximity graph in Example \ref{['ex1']}
• Figure 3.1: Modified arrowed proximity graph for $\mathcal{C}_0$ in Example \ref{['ex2']}
• Figure 3.2: Proximity graph of the configuration $\mathcal{C}_{\mathbb{P}^2}$ in Example \ref{['ex3']}

Theorems & Definitions (26)

• Conjecture 1.1: BNC
• Conjecture 1.2: WBNC
• Theorem 1
• Theorem 2
• Example 2.1
• Theorem 2.2
• Remark 2.3
• Theorem 2.4
• Definition 3.1
• Lemma 3.2