Nodal rational curves on Enriques surfaces of base change type

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.