Birational geometry: subgroups of the Cremona groups and their generators

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The decomposition group Dec(C) of a curve C, i.e. the subgroup of the Cremona group Bir(P^2) which preserve the curve C, is generated by quadratic elements in case C is a plane rational curve of degree 1,2 or 3. For every d which is at least 4, there is a plane rational curve C of degree d such that Dec(C) is not generated by its quadratic elements. This joint work with T. Ducat and S. Zimmermann has been accepted for publication in Mathematical Research Letters.In my joint work with A. Dubouloz and T. Kishimoto, we establish basic properties of Ga-threefolds whose algebraic quotient morphism is of a particularly simple form. Here Ga denotes the additive group over the base field, and a Ga-threefold is a variety of dimension 3 with a Ga-action. In particular we give a complete classification of the subclass of Ga-threefolds consisting of threefolds X endowed with proper Ga-actions, whose algebraic quotient morphisms are surjective with degenerate fibres isomorphic to the affine plane A^2 when equipped with their reduced structures. This work has been submitted to the journal Annali della Scuola Normale Superiore di Pisa (on October 2, 2017), and is currently under review.We (Heden and Mukai) studied the decomposition group of 5 lines in the projective plane and found 15 quadratic transformations in the group. I (Heden) later found new ones. By this discovery the solution becomes much harder than the case of 6 lines, for which Mukai determined the decomposition group completely.