Curves with low Harbourne constants on Kummer and Abelian surfaces

TL;DR

This paper constructs specific curves with low Harbourne constants on abelian and K3 surfaces, using configurations on Jacobian and Abelian surfaces to find examples with H-constants less than -4.

Contribution

It introduces new methods to construct curves with low Harbourne constants on Kummer and Abelian surfaces, expanding understanding of their geometric properties.

Findings

Curves with H-constants less than -4 on generic Jacobian K3 surfaces.

Use of Kummer configurations to produce low H-constant curves.

Examples of high-genus curve configurations on abelian surfaces.

Abstract

We construct and study curves with low H-constants on abelian and K3 surfaces. Using the Kummer $(16_{6})$-configurations on Jacobian surfaces and some $(16_{10})$-configurations of curves on $(1,3)$-polarized Abelian surfaces, we obtain examples of configurations of curves of genus $>1$ on a generic Jacobian K3 surface with H-constants $<-4$.