Bounded negativity, Miyaoka-Sakai inequality and elliptic curve configurations

Abstract

Similarly to the linear Harbourne constant recently defined, we study the elliptic $H$-constants of $\mathbb{P}^{2}$ and Abelian surfaces. We exhibit configurations of smooth plane cubic curves whose Harbourne index is arbitrarily close to $-4$. As a Corollary, we obtain that the global $H$-constant of any surface $X$ is less or equal to $-4$. Related to these problems, we moreover give a new inequality for the number and multiplicities of singularities of elliptic curves arrangements on Abelian surfaces, inequality which has a close similarity to the one of Hirzebruch for arrangements of lines in the plane.