Extremal rational elliptic threefolds (Q621290)

Let \(X\) be the blow-up of the base locus of a net of quadrics in \(\mathbb P^3\). Then \(X\) admits a natural elliptic fibration \(f:X\to \mathbb P^2\) with section. In this paper the author classifies nets of quadrics such that the Mordell-Weil group is finite. Such nets of quadrics turn out to contain several one-dimensional families reducible quadrics. Moreover, the discriminant curve of such a net (which is a quartic curve in \(\mathbb P^2\)) turns out to be a singular curve such that the configuration of singularities is one of the following \(E_7\), \(A_7\), \(D_6+A_1\), \(A_5+A_2\), \(D_4+3A_1\) or \(2A_3+A_1\).