Advanced Studies in Pure Mathematics

Existence of crepant resolutions

Toshihiro Hayashi, Yukari Ito, and Yuhi Sekiya

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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

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

Received: 10 July 2013
Revised: 22 October 2013
First available in Project Euclid: 23 October 2018

Permanent link to this document euclid.aspm/1540319488

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

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


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.

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