Linear weighted bounded negativity

TL;DR

This paper introduces a linear version of the weighted bounded negativity conjecture for smooth projective surfaces, establishing bounds on curve divisors and exploring cases for complex and rational surfaces.

Contribution

It formulates a new linear bounded negativity conjecture and proves existence of bounds in the complex case and for rational surfaces, using foliation techniques.

Findings

Existence of a common lower bound for $C^2/(D\,C)$ on complex surfaces.

Explicit bounds provided for rational surfaces.

Results are largely independent of the foliation used in proofs.

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 $\mathcal{F}$.