## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Higher Dimensional Algebraic Geometry: In honour of Professor Yujiro Kawamata's sixtieth birthday, K. Oguiso, C. Birkar, S. Ishii and S. Takayama, eds. (Tokyo: Mathematical Society of Japan, 2017), 185 - 202

### Existence of crepant resolutions

Toshihiro Hayashi, Yukari Ito, and Yuhi Sekiya

#### Abstract

Let $G$ be a finite subgroup of $\operatorname{SL}(n,\mathbb{C})$, then the quotient $\mathbb{C}^n/G$ has a Gorenstein canonical singularity. If $n=2\ \text{or}\ 3$, it is known that there exist crepant resolutions of the quotient singularity. In higher dimension, there are many results which assume existence of crepant resolutions. However, few examples of crepant resolutions are known. In this paper, we will show several trials to obtain crepant resolutions and give a conjecture on existence of crepant resolutions.

#### Article information

**Dates**

Received: 10 July 2013

Revised: 22 October 2013

First available in Project Euclid:
23 October 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1540319488

**Digital Object Identifier**

doi:10.2969/aspm/07410185

**Mathematical Reviews number (MathSciNet)**

MR3791214

**Zentralblatt MATH identifier**

1388.14051

**Subjects**

Primary: 14C05: Parametrization (Chow and Hilbert schemes) 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17] 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)

Secondary: 14E15: Global theory and resolution of singularities [See also 14B05, 32S20, 32S45] 14M25: Toric varieties, Newton polyhedra [See also 52B20]

**Keywords**

Quotient singularity McKay correspondence Hilbert scheme Crepant reosltuion Grobner basis toric variety

#### Citation

Hayashi, Toshihiro; Ito, Yukari; Sekiya, Yuhi. Existence of crepant resolutions. Higher Dimensional Algebraic Geometry: In honour of Professor Yujiro Kawamata's sixtieth birthday, 185--202, Mathematical Society of Japan, Tokyo, Japan, 2017. doi:10.2969/aspm/07410185. https://projecteuclid.org/euclid.aspm/1540319488