Nodal rational curves on Enriques surfaces of base change type

Using lattice theory, Hulek and Sch\"utt 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.