Bounded Negativity and Arrangements of Lines

TL;DR

This paper investigates the Bounded Negativity Conjecture for smooth complex surfaces, introduces constants measuring lower bound variance under birational transformations, and relates these to line arrangements on rational surfaces.

Contribution

It introduces constants $H(X)$ to quantify lower bound variance and connects these to line arrangements on rational surfaces, advancing understanding of the conjecture.

Findings

Defined constants $H(X)$ for measuring lower bound variance

Related $H({f P}^2)$ to line arrangements

Main theorem links negativity bounds to arrangements

Abstract

The Bounded Negativity Conjecture predicts that for any smooth complex surface $X$ there exists a lower bound for the selfintersection of reduced divisors on $X$. This conjecture is open. It is also not known if the existence of such a lower bound is invariant in the birational equivalence class of $X$. In the present note we introduce certain constants $H(X)$ which measure in effect the variance of the lower bounds in the birational equivalence class of $X$. We focus on rational surfaces and relate the value of $H({\mathbb P}^2)$ to certain line arrangements. Our main result is Theorem 3.3 and the main open challenge is Problem 3.10.