2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/144451Let $G$ be a finite abelian subgroup of $PGL(r-1,K)=\mathrm{Aut}(¶^{r-1}_K)$. 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 figuresAlgebraic GeometryCommutative Algebra13P10; 14L30; 14E15; 14M25text