# Hokkaido Mathematical Journal

## Hokkaido Mathematical Journal, 38 (2009) pp.249-265

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### Abstract

Let $G$ be a finite abelian subgroup of $PGL(r1,K)=\mathrm{Aut}(\P^{r-1}_K)$. In this paper, we prove that the normalization of the $G$-Hilbert scheme$\Hilb^G(\P^{r-1})$ is described as a toric variety, which corresponds to the Gr\"obner fan for some homogeneous ideal $I$ of $K[x_1, \ldots ,x_r]$.

MSC(Primary) 13P10 14L30, 14E15, 14M25 Gr\"obner fan, G-Hilbert schemes, toric singularity.