Singular rational curves on Enriques surfaces

TL;DR

This paper demonstrates that for specific integers, very general Enriques surfaces contain rational curves with particular genus and invariant properties, expanding understanding of their geometric structure.

Contribution

It establishes the existence of rational curves with prescribed genus and invariant on very general Enriques surfaces for certain integers.

Findings

Existence of rational curves with genus k on Enriques surfaces for k ≡ 1 mod 4

Construction of such curves with φ-invariant equal to 2

Results applicable to very general Enriques surfaces

Abstract

We show that for every $k\in\mathbb{Z}_+$, with $k\equiv_4 1$, the very general Enriques surface admits rational curves of arithmetic genus $k$ with $\phi$-invariant equal to 2.