On birational rigidity of singular del Pezzo fibrations of degree 1

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.

20 pages, Discussion on subtle differences of definitions of birational rigidity is added