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We prove that the group of birational transformations of a Del Pezzofibration of degree 3 over a curve is not simple, by giving a surjective grouphomomorphism to a free product of infinitely many groups of order 2. As aconsequence we also obtain that the Cremona group of rank 3 is not generated bybirational maps preserving a rational fibration. Besides, its subgroupgenerated by all connected algebraic subgroups is a proper normal subgroup.

Quotients of groups of birational transformations of cubic Del Pezzo fibrations