Bounded negativity, Miyaoka-Sakai inequality, and elliptic curve configurations. (English) Zbl 1405.14018

Roulleau, Xavier

Int. Math. Res. Not. 2017, No. 8, 2480-2496 (2017).

Summary: Similar to the linear Harbourne constant recently introduced in [T. Bauer et al., ibid. 2015, No. 19, 9456–9471 (2015; Zbl 1330.14007)], we study the elliptic \(H\)-constants of \(\mathbb P^2\) and of abelian surfaces. We also study the Harbourne indices of curves on these surfaces. In particular, we show that there are configurations of smooth plane cubic curves whose Harbourne indices are arbitrarily close to \(-4\). Consequently, we obtain that the \(H\)-constant of any surface \(X\) is less than 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 lines arrangements on the plane.

MSC:

14C17	Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14C20	Divisors, linear systems, invertible sheaves
14N20	Configurations and arrangements of linear subspaces
14K12	Subvarieties of abelian varieties