| 研究実績の概要 |
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.
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