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

TL;DR

This paper studies the non-rationality of divisors on 3-fold log Fano fibrations, establishing bounds on the gonality and genus of certain contracted divisors based on their coefficients.

Contribution

It provides new bounds on the gonality and genus of divisors contracted to points in 3-fold Fano fibrations, linking divisor coefficients to geometric properties.

Findings

Divisors with coefficient ≥ t are birational to P^1 × C.

Gonality of C depends only on t.

Genus of C is bounded if t > 1/2.

Abstract

In this paper we investigate non-rationality of divisors on 3-fold log Fano fibrations $(X,B)\to Z$ under mild conditions. We show that if $D$ is a component of $B$ with coefficient $\ge t>0$ 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 with gonality bounded depending only on $t$. Moreover, if $t>\frac{1}{2}$, then genus of $C$ is bounded depending only on $t$.