Birational geometry of del Pezzo fibrations with terminal quotient singularities

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 ${\mathrm{PSL}}_2(7)$ into the Cremona group.

arXiv commentIn this version some proofs of well known facts have been omitted

References in corpus (9)

• On G-Fano threefolds
• Simple finite subgroups of the Cremona group of rank 3
• Normal subgroups in the Cremona group (long version)
• Birational geometry of Fano direct products
• G-Fano threefolds, I
• Topologies and structures of the Cremona groups
• G-Fano threefolds, II
• Birational properties of pencils of del Pezzo surfaces of degree 1 and 2
• Birational Mori fiber structures of Q-Fano 3-fold weighted complete intersections, II

Cited by in corpus (1)

• On birational rigidity of singular del Pezzo fibrations of degree 1