When a quotient singularity is toric? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T03:24:59Z http://mathoverflow.net/feeds/question/89533 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89533/when-a-quotient-singularity-is-toric When a quotient singularity is toric? Mohammad F.Tehrani 2012-02-25T23:50:38Z 2012-02-26T00:18:04Z <p>Let $G \subset SL(n,\mathbb{C})$ be a cyclic subgroup of finite order, Is it true that $\mathbb{C}^n /G$ is toric ? If not then when it is ?</p> http://mathoverflow.net/questions/89533/when-a-quotient-singularity-is-toric/89534#89534 Answer by David Speyer for When a quotient singularity is toric? David Speyer 2012-02-26T00:18:04Z 2012-02-26T00:18:04Z <p>Yes, it is true. Let $G \cong \mathbb{Z}/m$ act by $\mathrm{diag}(\zeta^{a_1}, \zeta^{a_2}, \ldots, \zeta^{a_n})$, where $\zeta$ is a primitive $n$-th root of unity. Let $S$ be the semigroup $\{ (b_1, \ldots, b_n) \in \mathbb{Z}_{\geq 0}^n : \sum a_i b_i \equiv 0 \mod m \}$. Then the quotient is Spec of the semigroup ring $\mathbb{C}[S]$. Since the semigroup is torsion free, finitely generated and saturated, the corresponding Spec is an affine toric variety.</p>