Hilbert schemes of finite abelian group orbits and Grobner fans

Abstract

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

13 pages, 3 figures