2013 年 65 巻 1 号 p. 105-130
We propose a method to compute a desingularization of a normal affine variety $X$ endowed with a torus action in terms of a combinatorial description of such a variety due to Altmann and Hausen. This desingularization allows us to study the structure of the singularities of $X$. In particular, we give criteria for $X$ to have only rational, ($\boldsymbol{Q}$-)factorial, or ($\boldsymbol{Q}$-)Gorenstein singularities. We also give partial criteria for $X$ to be Cohen-Macaulay or log-terminal. Finally, we provide a method to construct factorial affine varieties with a torus action. This leads to a full classification of such varieties in the case where the action is of complexity one.
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