On Conjugacy Classes of the Klein Simple Group in Cremona Group

Abstract

We consider countably many three dimensional PSL2(F7)-del Pezzo surface fibrations over P 1 . Conjecturally they are all irrational except two families, one of which is the product of a del Pezzo surface with P 1 . We show that the other model is PSL2(F7)-equivariantly birational to P 2 × P 1 . Based on a result of Prokhorov, we show that they are non-conjugate as subgroups of the Cremona group Cr3(C).2010 Mathematics Subject Classification. 14E07.

Birational geometry of del Pezzo fibrations with terminal quotient singularities

“…It is expected that del Pezzo fibrations of degree 1 also have models with good singularities, these models have to be 6-Gorenstein. Therefore for del Pezzo fibrations of degree 1 it is natural to consider 1 2 (1, 1, 1) and 1 3 (1, 1, 2)-singularities [19]. The following is a well-known and widely believed conjecture.…”

“…Note that there are difficulties when F contains the 1 2 (1, 1, 1)-point. There will be 'half-line': curves of degree 1 2 . Thus the bound on multiplicity becomes mult P Z v 8γn 2 and it no longer contradicts Corti inequality.…”

“…The good model of del Pezzo fibrations of degree 2 are 2-Gorenstein. Hence good models may not admit smoothing since the singularities of 2-Gorenstein varieties are degenerations of the 1 2 (1, 1, 1)-singularity which does not admit smoothing. In particular it means that smooth del Pezzo fibrations are not dense in moduli.…”

“…It is easy to see that X n admits the action of the group PSL 2 (7) because right-hand side of the equation of X n is a polynomial defining the Klein quartic. Variety X n has 2n points of the type 1 2 (1, 1, 1) and has no other singularities. Thus X n is terminal and Q-factorial.…”

“…Theorem 1.1. Let π : X → P 1 be a del Pezzo fibration of degree 2 with only 1 2 (1, 1, 1)-singularities (simplest terminal quotient singularities). Suppose Pic(X) = Z ⊕ Z and suppose X satisfies K 2condition, that is K 2 X is not in the interior of the Mori cone NE(X).…”