Singular rational curves on Enriques surfaces

TL;DR

The paper addresses the problem of realizing rational curves of prescribed genus on very general Enriques surfaces. It combines Horikawa's double-cover model of K3s over quadrics, the theory of logarithmic Severi varieties, and regeneration theorems for curves on K3s to transfer constructions from Enriques surfaces of base-change type to the very general case. The main result shows that for every $k$ with $k \equiv_4 1$, there exist rational curves of arithmetic genus $k$ on the very general Enriques surface with $\phi=2$, expanding the landscape beyond previously known nodal and base-change examples. The work leverages deformation theory and lattice-theoretic constraints to prove the density of such curves in moduli, highlighting the intricate interplay between Enriques geometry, K3 covers, and Severi-type parameter spaces. This provides a concrete, broad class of rational curves on very general Enriques surfaces and clarifies the possible genera compatible with the $\phi$-invariant constraint.

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 $φ$-invariant equal to 2.

Theorems & Definitions (10)

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• proof : Proof of Theorem \ref{['main']}