Hilbert schemes of finite abelian group orbits and Grobner fans

Abstract

Let $G$ be a finite abelian subgroup of $PGL(r-1,K)=\mathrm{Aut}(\P^{r-1}_K)$. In this paper, we prove that the normalization of the $G$-orbit 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 $K[x_1, ..., x_r]$.