Curves on Enriques surfaces and on rational elliptic surfaces

We explore the geometry of the Enriques surfaces and of the rational elliptic surfaces, particularly focusing on the genus 1 pencils they admit, on the Severi varieties of curves on them and on the rational curves lying on them. We investigate the open problem of the existence of rational curves in the very general Enriques surface: exploiting the "regeneration" results due to Chen, Gounelas and Liedtke about curves on K3 surfaces and a construction of some particular Enriques surfaces by Hulek and Schütt, we prove that for every $k\equiv_4 1$, the very general Enriques surface admits rational curves of arithmetic genus $k$ and $\phi=2$.