Bounded Negativity and Arrangements of Lines

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.

v2, rewritten, extra material on arrangements of real lines, to appear in International Mathematics Research Notices

References in corpus (1)

• Bounded negativity of self-intersection numbers of Shimura curves on Shimura surfaces

Cited by in corpus (14)

• On the containment problem
• Bounded negativity, Harbourne constants and transversal arrangements of curves
• Symbolic powers of planar point configurations II
• The 21 reducible polars of Klein's quartic
• Harbourne constants and conic configurations on the projective plane
• Line arrangements with the maximal number of triple points
• From Pappus Theorem to parameter spaces of some extremal line point configurations and applications
• Harbourne constants, pull-back clusters and ramified morphisms
• Local negativity of surfaces with non-negative Koidara dimension and transversal configurations of curves
• On absolute linear Harbourne constants
• On the degree of curves with prescribed multiplicities and bounded negativity
• Line arrangements with many triple points
• Towards the weighted bounded negativity conjecture for blow-ups of algebraic surfaces
• The cone of curves and the Cox ring of rational surfaces over Hirzebruch surfaces