Nodal rational curves on Enriques surfaces of base change type
Simone Pesatori
TL;DR
The paper provides a lattice‑free geometric construction of Enriques surfaces of base change type and proves that the associated rational bisections are generically nodal. It shows that for each $m\ge 0$ the very general Enriques surface in the $m$‑special family carries a countable set of nodal rational curves with a genus‑dependent formula, forming a rank‑8 subgroup of the automorphism group. The work connects the geometry of Enriques surfaces to rational elliptic surfaces via the Mordell–Weil group, and develops logarithmic Severi varieties to control tangency conditions, enabling explicit computations of torsion multisections and nodal curves. The results advance understanding of nodal rational curves on Enriques surfaces and provide tools for explicit automorphism‑group and genus computations on these surfaces.
Abstract
Using lattice theory, Hulek and Schütt proved that for every $m\in\mathbb{Z}_+$ there exists a nine-dimensional family $\mathcal{F}_m$ of K3 surfaces covering Enriques surfaces having an elliptic pencil with a rational bisection of arithmetic genus $m$. We present a purely geometrical lattice free construction of these surfaces, that allows us to prove that generically the mentioned bisections are nodal. Moreover, we show that, for every $m\in\mathbb{Z}_+$, the very general Enriques surface covered by a K3 surface in $\mathcal{F}_m$ admits a countable set of nodal rational curves of arithmetic genus $(4k^2-4k+1)m+4k^2-4k$ for every $k\in\mathbb{Z}_+$, that form a rank 8 subgroup of the automorphism group of the surface. As an application, we compute the linear class of the $n$-torsion multisection for every $n\in\mathbb{N}$ for a general rational elliptic surface.