Birational geometry of del Pezzo fibrations with terminal quotient singularities

TL;DR

This paper proves birational rigidity for certain singular del Pezzo fibrations, establishing their non-rationality and exploring implications for embeddings into the Cremona group.

Contribution

It introduces new birational rigidity results for del Pezzo fibrations with specific singularities, extending understanding beyond smooth cases.

Findings

Birational rigidity established for del Pezzo fibrations with simple non-Gorenstein singularities.

Non-rationality of these fibrations proven under the $K^2$-condition.

Applications to embeddings of $ ext{PSL}_2(7)$ into the Cremona group.

Abstract

Del Pezzo fibrations appear as minimal models of rationally connected varieties. The rationality of smooth del Pezzo fibrations is a well studied question but smooth fibrations are not dense in moduli. Little is known about the rationality of the singular models. We prove birational rigidity, hence non-rationality, of del Pezzo fibrations with simple non-Gorenstein singularities satisfying the famous $K^2$-condition. We then apply this result to study embeddings of $\operatorname{PSL}_2(7)$ into the Cremona group.