The author approaches the existence problem of a dihedral cover of Hirzebruch surfaces \(\Sigma_d\) with prescribed branch curves via the theory of elliptic surfaces and Mordell-Weil lattices. The elliptic surface \(S\) appears as a double cover of \(\Sigma_d\) and the potential branch curves on \(S\) are produced as sections of \(S/\mathbb{P}^1\) by the group operations on the Mordell-Weil groups.
The setting is as follows. Let \(\pi: X\rightarrow Y\) be a Galois cover with \(\mathrm{Gal}(\pi)\simeq D_{2p}\), the dihedral group of order \(2p\). The cover \(\pi\) is called elliptic if (i) the canonical double cover \(f: D(X/Y)\rightarrow Y\) corresponding to the index two subgroup of \(\mathrm{Gal}(\pi)\) has a structure of a relatively minimal elliptic surface \(D(X/Y)=S\rightarrow \mathbb{P}^1\) with a section, and (ii) the covering transformation \(\sigma_f\) of \(f\) coincides with the fiberwise inversion of the elliptic surface \(S\). Note that then \(Y\) is ruled over \(\mathbb{P}^1\), hence is a blowup \(\widehat{\Sigma_d}\) of a Hirzebruch surface \(\Sigma_d\) for some even \(d\).
Under these notation, Theorem 3.3 is concisely as follows. Let \(p\) be an odd prime which does not divide the order of the torsion subgroup of \(\mathrm{MW}(S)\). Let \(s_1,s_2\in \mathrm{MW}(S)\) be sections such that \(s_i\) are not \(p\)-divisible in the Mordell-Weil group \(\mathrm{MW}(S)\). Then there exists a \(D_{2p}\)-cover \(X_{p}\rightarrow \widehat{\Sigma_d}\) such that the horizontal part of the branch of \(X_p/S\) is \(s_1+s_2+\sigma_f(s_1+s_2)\) if and only if the images \(\overline{s}_i\in \mathrm{MW}(S)\otimes \mathbb{Z}/p\mathbb{Z}\) are linearly dependent.
As an application, the author considers the case \(S\) is rational and constructs a Zariski pair of conic-line arrangement of degree \(7\).