On birational rigidity of singular del Pezzo fibrations of degree 1

TL;DR

This paper establishes a sufficient condition for the birational superrigidity of degree 1 del Pezzo fibrations with specific singularities, extending previous criteria and applying it to fibrations embedded in toric bundles.

Contribution

It generalizes the $K^2$-condition for birational superrigidity to certain singular del Pezzo fibrations of degree 1.

Findings

A new sufficient condition for birational superrigidity of these fibrations.

Equivalence of superrigidity and the $K$-condition for fibrations in toric bundles.

Extension of previous rigidity results to singular cases with specific singularities.

Abstract

We give a sufficient condition for birational superrigidity of del Pezzo fibrations of degree $1$ with only $\frac{1}{2} (1,1,1)$ singular points, generalizing the so called $K^2$-condition. As an application, we also prove that a del Pezzo fibrations of degree $1$ with only $\frac{1}{2} (1,1,1)$ singular points embedded in a toric $\mathbb{P} (1,1,2,3)$-bundle over $\mathbb{P}^1$ is birationally superrigid if and only if it satisfies the $K$-condition.