Tohoku Mathematical Journal
2013
March
SECOND SERIES VOL. 65, NO. 1

Tohoku Math. J.
65 (2013), 105-130

Title NORMAL SINGULARITIES WITH TORUS ACTIONS

Author Alvaro Liendo and Hendrik Süss

(Received October 6, 2010, revised July 31, 2012)
Abstract. 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.

2000 Mathematics Subject Classification. Primary 14J17; Secondary 14E15.
Key words and phrases. Torus actions, T-varieties, characterization of singularities, toroidal desingularization.

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