Bounded negativity, Miyaoka-Sakai inequality and elliptic curve configurations

TL;DR

This paper investigates elliptic curve configurations on surfaces, establishing bounds on their negativity indices and introducing new inequalities related to singularities, extending concepts like the Harbourne constant to Abelian surfaces.

Contribution

It introduces elliptic H-constants for surfaces, constructs configurations with Harbourne index near -4, and derives a new inequality for singularities of elliptic curve arrangements.

Findings

Configurations with Harbourne index close to -4

Global H-constant of any surface ≤ -4

New inequality for singularities of elliptic arrangements

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.