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Similarly to the linear Harbourne constant recently defined, we study theelliptic $H$-constants of $\mathbb{P}^{2}$ and Abelian surfaces. We exhibitconfigurations of smooth plane cubic curves whose Harbourne index isarbitrarily 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 theseproblems, we moreover give a new inequality for the number and multiplicitiesof singularities of elliptic curves arrangements on Abelian surfaces,inequality which has a close similarity to the one of Hirzebruch forarrangements of lines in the plane.

Bounded negativity, Miyaoka-Sakai inequality and elliptic curve configurations