Bounded negativity and arrangements of lines. (English) Zbl 1330.14007

Bauer, Thomas; Di Rocco, Sandra; Harbourne, Brian; Huizenga, Jack; Lundman, Anders; Pokora, Piotr; Szemberg, Tomasz

Int. Math. Res. Not. 2015, No. 19, 9456-9471 (2015).

It is conjectured that the self intersection of reduced curves in a smooth projective surface is lower bounded. Looking for evidences of such conjecture it is natural to ask if this boundeness property is of birational nature and moreover if the possible bounds corresponding to two birational surfaces are related. The paper under review deals with this last question in the case of rational surfaces. Since it is possible to construct curves such that \(C^2\) is arbitrarily negative just choosing properly \(s\) points in the plane to blow up, it is natural to try to control the ratio \(C^2/s\). The main result (Thm. 3.3) is that for reduced curves \(C\) which are strict transforms of configurations of lines on the blow up of the plane in \(s\)-points, it holds that \(C^2/s>-4\). The lack of examples leads the authors to propose the question (Problem 3.10) on the generality of the bound \(-4\) (not only for curves coming from configurations of lines).

Reviewer: Roberto Munoz (Madrid)

MSC:

14C17	Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14C20	Divisors, linear systems, invertible sheaves
14N20	Configurations and arrangements of linear subspaces

Keywords:

negative curves; smooth surfaces; birationality; arrangements of lines