Birational geometry: subgroups of the Cremona groups and their generators

当C是1次、2次或3次平面有理曲线时，曲线C的分解群DEC(C)，即保持曲线C的Cremona群Bir(P^2)的子群，是由二次元生成的.对于至少为4的任意d，存在d次平面有理曲线C，使得DEC(C)不是由它的二次元生成的.这项与T.Ducat和S.Zimmermann的联合工作已被接受发表在数学研究快报上。在我与A.DuBouloz和T.Kishimoto的联合工作中，我们建立了GA-三重的基本性质，其代数商态射具有特别简单的形式。这里，Ga表示基场上的加法基团，而Ga-三重是具有Ga-作用的维度3的变化。特别地，我们给出了由赋有适当Ga作用的三折X组成的Ga-三折的子类的完全分类，其代数商态射是满射的，当具有它们的约化结构时，它的简并纤维同构于仿射平面A^2。我们(Heden和Mukai)研究了射影平面上5条直线的分解群，发现了群中的15个二次变换。我(海登)后来发现了新的。通过这一发现，解变得比Mukai完全确定分解群的6行的情况困难得多。

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.