## When a quotient singularity is toric?

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

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 ?

-

1
The "cyclic" condition is redundant, the quotient is toric for any finite Abelian group. – Dmitri Feb 26 2012 at 0:21
I kind of knew that but I just simplified the problem. I could not remember any proof or procedure. How is the the construction? – Mohammad F.Tehrani Feb 26 2012 at 0:34

3
The action of any finite Abelian group on $\mathbb C^n$ is diagonalisable - this is just linear algebra. So the action commutes with diagonal $(\mathbb C^*)^n$, this makes your quotient toric. – Dmitri Feb 26 2012 at 1:26

## 1 Answer

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.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 11, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.897322952747345, "perplexity_flag": "middle"}