Mordell-Weil lattice of higher genus fibration on a Fermat surface (Q2788597)

The \textit{Fermat surface} \(X_m\) is the smooth surface in \(\mathbb P^{\#1}\) given by the equation \(x_0^m+x_1^m+x_2^m+x_3^m=0\). (The author mainly works over \(\mathbb C\), although the case of positive characteristic is also discussed occasionally.) Assume that \(m\geq 4\). The surface \(X_m\) contains \(3m^2\) lines, which are all projectively equivalent. A line \(l_0\subset X_m\) defines a fibration \(X_m\to\mathbb P^{\#1}\) of genus \(\frac12(m-2)(m-3)\): its fibers are the residual curves cut on \(X_m\) by the planes through \(l_0\). The author computes the Mordell-Weil lattice \(M\) of this fibration. In general, \(\mathrm{rk }M=\rho(X_m)-(4m-2)\), where \(\rho(X_m)\) is the Picard rank (see [\textit{T. Shioda}, J. Fac. Sci., Univ. Tokyo, Sect. I A 28, 725--734 (1981; Zbl 0567.14021)]). In the special case of \(m\) prime to \(6\) this simplifies to \(\mathrm{rk }M=(3m-5)(m-2)\) and \(M\) is generated by the classes of lines \(l\subset X_m\) disjoint from~\(l_0\). (Such lines are obviously sections of the fibration.) The height intersection pairing restricted to the classes of lines is also computed, but it is too complicated to be cited here.