Linear weighted bounded negativity

TL;DR

The paper develops a linear variant of the weighted bounded negativity conjecture (LWBNC) for smooth complex projective surfaces by bounding $\frac{C^2}{D\cdot C}$ for negative curves $C$ against big and nef divisors $D$ with $D\cdot C>0$. It proves a general, ray-based bound in the complex case and provides explicit, computable bounds for rational surfaces, leveraging foliations as a key tool though many results ultimately depend only on combinatorial data from infinitely near point configurations. The authors introduce and analyze the invariant $\nu_D(X)$ and the Delta-sets $\Delta(X;G,\epsilon)$, establishing finiteness and universal-type bounds away from limit rays, and then specialize to rational surfaces to obtain precise, configuration-driven constants. The work advances understanding of BNC variants by delivering explicit bounds in the rational setting and highlighting the role of foliations in deriving and organizing these bounds, with practical implications for computing negativity bounds from proximity graphs.

Abstract

We propose a linear version of the weighted bounded negativity conjecture. It considers a smooth projective surface $X$ over an algebraically closed field of characteristic zero and predicts the existence of a common lower bound on $C^2/(D\cdot C)$ for all reduced and irreducible curves $C$ and all big and nef divisors such that $D\cdot C>0$, both on $X$. We prove that, in the complex case, there exists such a bound for all nef divisors spanning a ray out an open covering of the limit rays of negative curves. In the same vein, we provide explicit bounds when $X$ is a rational surface. Our proofs involve the existence of a foliation $\mathcal{F}$ on $X$ but most of our results are independent of $\mathcal{F}$.

Figures (2)

• Figure 1: Explanatory picture of Theorem \ref{['thm_cota_paco']}.
• Figure 2: Proximity graph of $\mathcal{C}$.

Theorems & Definitions (40)

• Lemma 2.1
• Theorem 2.2
• Proposition 2.3
• Theorem 3.1
• proof
• Corollary 3.2
• Theorem 3.3
• proof
• Lemma 3.4
• proof