Bounded negativity, Harbourne constants and transversal arrangements of curves

Abstract

The Bounded Negativity Conjecture predicts that for every complex projective surface $X$ there exists a number $b(X)$ such that $C^2\ge -b(X)$ holds for all reduced curves $C\subset X$. For birational surfaces $f:Y\to X$ there have been introduced certain invariants (Harbourne constants) relating to the effect the numbers $b(X)$, $b(Y)$ and the complexity of the map $f$. These invariants have been studied previously when $f$ is the blowup of all singular points of an arrangement of lines in ${\mathbb{P}}^2$, of conics and of cubics. In the present note we extend these considerations to blowups of ${\mathbb{P}}^2$ at singular points of arrangements of curves of arbitrary degree $d$. We also considerably generalize and modify the approach witnessed so far and study transversal arrangements of sufficiently positive curves on arbitrary surfaces with the non-negative Kodaira dimension.

This is the final version, incorporating the suggestions of the referee, to appear in Annales de l'Institut Fourier Grenoble

References in corpus (3)

• Bounded Negativity and Arrangements of Lines
• Harbourne constants and conic configurations on the projective plane
• Bounded Negativity and Symplectic 4-Manifolds

Cited by in corpus (6)

• Conic-line arrangements in the complex projective plane
• The orbifold Langer-Miyaoka-Yau inequality and Hirzebruch-type inequalities
• Local negativity of surfaces with non-negative Koidara dimension and transversal configurations of curves
• Algebraic properties of Levi graphs associated with curve arrangements
• On the degree of curves with prescribed multiplicities and bounded negativity
• Curve configurations in the projective plane and their characteristic numbers