Hilbert schemes of finite abelian group orbits and Grobner fans

TL;DR

This paper demonstrates that the normalization of the G-orbit Hilbert scheme for a finite abelian subgroup of projective automorphisms is a toric variety associated with a Gr"obner fan, linking algebraic geometry and combinatorics.

Contribution

It establishes a toric description of the normalized G-orbit Hilbert scheme using Gr"obner fans for the first time in this context.

Findings

Normalized G-orbit Hilbert schemes are toric varieties.

These toric varieties correspond to specific Gr"obner fans.

Provides a new geometric-combinatorial link in algebraic geometry.

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 $K[x_1, ..., x_r]$.