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 K2‐condition. We then apply this result to study embeddings of prefixPSL2false(7false) into the Cremona group.

Families of embeddings of the alternating group of rank 5 into the Cremona group

“…The remaining cases are A 6 and PSL 2 (7). Some progress toward classification the embeddings of these groups has been made in [3] and [11], so far the results agree with the following expectation.…”

“…A 5 S 5 1, thus the statement follows from [2, Theorem 1.5] (see[11, Theorem 3.6] for G-invariant version).…”

The rationality problem for conic bundles

“…For results on birational rigidity we refer to [Isk95], [Puk98], [Puk13, Ch. 4, § 1], [Che05], [Sob02], [Gri00], [SC11], [Kry16]. A higher-dimensional generalization was discussed in a recent paper [Puk17].…”

On birational rigidity of singular del Pezzo fibrations of degree 1

“…In the context of Minimal Model Program, it is natural and important to study singular del Pezzo fibrations. Recently, there is some progress for singular del Pezzo fibrations of degree 2: Krylov [17] and Ahmadinezhad–Krylov [3] proved that a del Pezzo fibration of degree 2 with only singular points of type satisfying the ‐condition is birationally rigid under some additional assumptions.…”