Bounding non-rationality of divisors on 3-fold Fano fibrations

Abstract

Abstract In this paper we investigate non-rationality of divisors on 3-fold log Fano fibrations ( X , B ) → Z {(X,B)\to Z} under mild conditions. We show that if D is a component of B with coefficient ≥ t > 0 {\geq t>0} which is contracted to a point on Z, then D is birational to ℙ 1 × C {\mathbb{P}^{1}\times C} , where C is a smooth projective curve with gonality bounded depending only on t. Moreover, if t > 1 2 {t>\frac{1}{2}} , then genus of C is bounded depending only on t.