特異曲線へ収縮する森収縮写像の分類

在主要的过程中，他的研究，托马斯杜卡特遇到了一类特殊的代数品种称为集群品种。这些品种具有非常丰富的组合结构，并且可以根据根系的数据来定义。考虑到这些集群变种所享有的大量对称性，它们是用作关键变种的理想候选者。在与Stephen Coughlan的一个联合项目中，他们一直在使用这些簇中的一些簇来构建Q-Fano 3-folds的许多新例子，包括以前很难研究的情况（例如Q-Fano 3-folds X，其中反正则线性系统是空的）。他们预计将有许多其他应用的这种方法，例如，建设表面的一般类型。在一个单独的一块工作，他曾与艾萨克黑登和苏珊娜齐默尔曼的主题分解组的平面二次曲线和平面理性三次。平面曲线的分解群是由平面的双有理映射给出的平面Cremona群的子群，该双有理映射限制于曲线的双有理映射。继从他们以前的工作，他们能够给一个完整的描述这些分解群的平面理性曲线的程度最多3。

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.