Bounding non-rationality of divisors on 3-fold Fano fibrations (Q2238197)

In this paper the authors study the non-rationality of divisors on 3-fold log Fano fibrations \((X,B)\to Z \) under mild conditions. By assumption \((X,B)\) is klt and \(-(K_X+B)\) is ample over \(Z\). In particular they show that if \(D\) is a component of \(B\) with coefficient \(t > 0\) which is contracted to a point on \(Z\), then D is birational to \(\mathbb P ^1\times C\) , where \(C\) is a smooth projective curve whose gonality bounded depending only on \(t\). Moreover, if \(t > \frac 1 2\), then the genus of \(C\) is bounded depending only on \(t\).