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Register a new userModular quotient varieties and singularities by the cyclic group of order $2p$
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We classify all $n$-dimensional reduced Cohen-Macaulay modular quotient variety $mathbb{A}_mathbb{F}^n/C_{2p}$ and study their singularities, where $p$ is a prime number and $C_{2p}$ denotes the cyclic group of order $2p$. In particular, we present an example that demonstrates that the problem proposed by Yasuda cite[Problem 6.6]{Yas2015} has a negative answer if the condition that $G$ is a small subgroup was dropped.
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Let $Gsubseteq GL(n)$ be a finite group without pseudo-reflections. We present an algorithm to compute and verify a candidate for the Cox ring of a resolution $Xrightarrow mathbb{C}^n/G$, which is based just on the geometry of the singularity $mathbb{C}^n/G$, without further knowledge of its resolutions. We explain the use of our implementation of the algorithms in Singular. As an application, we determine the Cox rings of resolutions $Xrightarrow mathbb{C}^3/G$ for all $Gsubseteq GL(3)$ with the aforementioned property and of order $|G|leq 12$. We also provide examples in dimension 4.
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