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

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.

Anti-pluricanonical systems on Fano varieties

“…In higher dimensions, the degree of irrationality irr(D) of a variety D is the least possible degree of dominant rational maps D P dim D (see §2.16). The first named author and Loginov ask in [BL21] whether a similar result to Theorem 1.1 for klt Fano fibrations of higher dimensions should also hold.…”

“…In the paper [BL21], the first named author and Loginov study the non-rationality property of degenerations of klt del Pezzo surfaces and prove how far the components of the degenerations can be from being rational. More precisely, the boundedness of irrationality of degenerations is proved in [BL21] as follows. In particular, by [BL21] Theorem 1.1, the Date: January 17, 2024.…”

“…More precisely, the boundedness of irrationality of degenerations is proved in [BL21] as follows. In particular, by [BL21] Theorem 1.1, the Date: January 17, 2024. 2010 MSC: 14J45, 14E30, 14C20, 14D06.…”

“…Note that the genus g(C) can be arbitrarily large if t ≤ 1 2 (see [BL21] Example 2.3). Moreover, Theorem 1.1 follows from a more general result for Fano-type log Calabi-Yau fibrations where the base Z of the fibration can have dimension ≥ 1 (see [BL21] Theorem 1.3). Recall that the gonality gon(C) of a smooth curve C is the smallest degree of the dominant morphisms of curves C → P 1 .…”

Singularities on toric fibrations

“…The main point is that, putting z = f (S), by Theorem 1.5, (X, tf * z) is klt for some fixed t > 0. This allows us to apply the results of [17]. In upcoming work we bound the genus of C and treat similar statements in higher dimension.…”