Families of curves over \(\mathbb P^1\) with 3 singular fibers (Q357420)

Nontrivial semistable families of curves with minimal number of singular fibers have remarkable arithmetic and geometric properties. In the paper under review, the authors consider a relatively minimal genus \(g\) fibration \(S\) over \(\mathbb P^1\) with \(3\) singular fibers, \(2\) of which are semistable. They prove that \(S\) is a rational surface, the Mordell-Weil group is finite, and \(2g\leq -K_S^2 \leq 4g-4\). For \(g=2^n\), they construct examples to show that such fibrations exist.