Birational geometry: subgroups of the Cremona groups and their generators

曲线C的分解群Dec(C)，即保存曲线C的Cremona群Bir（P^2）的子群，当C是1次、2次或3次的平面有理曲线时，由二次元生成。对于每一个至少为4的d，存在一个d次的平面有理曲线C，使得Dec(C)不是由它的二次元生成的。这个与T.杜卡特和S.齐默尔曼的联合工作已经被接受发表在数学研究快报上。在我与a . Dubouloz和T. Kishimoto的合作工作中，我们建立了代数商态射具有特别简单形式的ga -三折子的基本性质。这里Ga表示基域上的加性群，Ga-三重是具有Ga-作用的维数3的变化。特别地，我们给出了由具有适当ga -作用的三倍X组成的ga -三倍子类的完全分类，这些三倍子类的代数商态是满射的，当它们具有它们的约化结构时，它们的简并纤维同构于仿射平面a ^2。这项工作已提交给Annali della Scuola Normale Superiore di Pisa杂志（2017年10月2日），目前正在审查中。我们（Heden和Mukai）研究了5条线在射影平面上的分解群，并在群中找到了15个二次变换。我（海登）后来又找到了新的。通过这一发现，解决方案比Mukai完全确定分解群的6条线的情况要困难得多。

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.