The T-graph of a multigraded Hilbert scheme

TL;DR

This paper studies the T-graph of multigraded Hilbert schemes, providing combinatorial conditions for edges and explicit descriptions for the Hilbert scheme of points in the plane, revealing dependence on the ground field.

Contribution

It introduces a combinatorial necessary condition for edges in the T-graph and describes the equations for one-dimensional T-orbit schemes in the plane case, also showing field dependence.

Findings

Provides a combinatorial necessary condition for T-graph edges.

Explicitly describes equations for one-dimensional T-orbit schemes in the plane.

Shows the T-graph depends on the ground field.

Abstract

The T-graph of a multigraded Hilbert scheme records the zero and one-dimensional orbits of the T = (K^*)^n action on the Hilbert scheme induced from the T-action on A^n. It has vertices the T-fixed points, and edges the one-dimensional T-orbits. We give a combinatorial necessary condition for the existence of an edge between two vertices in this graph. For the Hilbert scheme of points in the plane, we give an explicit combinatorial description of the equations defining the scheme parameterizing all one-dimensional torus orbits whose closures contain two given monomial ideals. For this Hilbert scheme we show that the T-graph depends on the ground field, resolving a question of Altmann and Sturmfels.