特異曲線へ収縮する森収縮写像の分類

在他的主要研究过程中，托马斯·杜卡特遇到了一类特殊的代数变种，称为簇变种。这些品种具有非常丰富的组合结构，可以根据根系的数据进行定义。鉴于这些集群品种具有很大的对称性，它们是用作关键品种的理想候选品种。在与Stephen Coughlan的一个联合项目中，他们一直在使用其中的一些簇簇来构造Q-Fano 3-折叠的许多新例子，包括以前很难研究的情况(例如反正则线性系统为空的Q-Fano 3-折叠X)。他们希望这种方法还有很多其他的应用，例如构造一般类型的曲面。在另一项单独的工作中，他与Isac Heden和Susanna Zimmermann合作研究了平面二次曲线和平面有理三次曲线的分解群的主题。平面曲线的分解群是平面的Cremona群的子群，它由平面的双调映射给出，限制于该曲线的双调映射。在前人工作的基础上，他们能够给出最多3次的平面有理曲线的这些分解群的完整描述。

During the main course of his research, Thomas Ducat has come across a special class of algebraic varieties called cluster varieties. These varieties have a very rich combinatorial structure and can be defined in terms of the data of a root system. Given the large amount of symmetry that these cluster varieties enjoy, they are ideal to candidates to be used as key varieties. In a joint project with Stephen Coughlan, they have been using some of these cluster varieties to construct many new examples of Q-Fano 3-folds, including cases that were previously very difficult to study (such as Q-Fano 3-folds X for which the anticanonical linear system is empty). They expect there will be many other applications of this method, e.g. constructing surfaces of general type.In a separate piece of work, he has collaborated with Isac Heden and Susanna Zimmermann on the topic of the decomposition groups of plane conics and plane rational cubics. The decomposition group of a plane curve is the subgroup of the plane Cremona group given by birational maps of the plane which restrict to a birational map of the curve. Following on from their previous work they were able to give a complete description of these decomposition groups for plane rational curves of degree at most 3.