Curves with low Harbourne constants on Kummer and 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 $$<-~4$$ .