On non-rigid del Pezzo fibrations of low degree

TL;DR

This paper studies non-rigid del Pezzo fibrations of low degree, classifying certain hypersurfaces in specific bundles over projective spaces that exhibit non-trivial Sarkisov links, expanding understanding of their birational geometry.

Contribution

It classifies hypersurfaces in $P(1,1,1,2)$ bundles over $P^1$ with non-trivial Sarkisov links and extends results to cubic surface fibrations over $P^2$.

Findings

Classified hypersurfaces with non-trivial Sarkisov links in specific del Pezzo fibrations.

Identified conditions under which these fibrations change Mori fibre space structure.

Extended classification to cubic surface fibrations over $P^2$.

Abstract

We consider $\mathbb{P}(1,1,1,2)$ bundles over $\mathbb{P}^1$ and construct hypersurfaces of these bundles which form a degree 2 del Pezzo fibration over $\mathbb{P}^1$ as a Mori fibre space. We classify all such hypersurfaces whose type $\III$ or $\IV$ Sarkisov links pass to a different Mori fibre space. A similar result for cubic surface fibrations over $\mathbb{P}^2$ is also presented.