Graphs of Schemes Associated to Group Actions

TL;DR

This paper studies the connectivity properties of graphs derived from algebraic schemes with torus actions, establishing a link between the connectedness of these graphs and the schemes themselves, with applications to Hilbert schemes.

Contribution

It introduces the $A$-graph for schemes with torus actions and proves its connectedness characterizes the scheme's connectedness, applying this to Hilbert schemes.

Findings

The $A$-graph of a scheme is connected iff the scheme is connected.

The $A$-graph provides a new combinatorial tool for studying scheme connectedness.

Application to Hartshorne's theorem on Hilbert scheme connectedness.

Abstract

Let $X$ be a proper algebraic scheme over an algebraically closed field. We assume that a torus $T$ acts on $X$ such that the action has isolated fixed points. The $T$-graph of $X$ can be defined using the fixed points and the one dimensional orbits of the $T$-action. If the upper Borel subgroup of the general linear group with maximal torus $T$ acts on $X$, then we can define a second graph associated to $X$, called the $A$-graph of $X$. We prove that the $A$-graph of $X$ is connected if and only if $X$ is connected. We use this result to give a proof of Hartshorne's theorem on the connectedness of Hilbert scheme in the case of $d$ points in $\mathbb{P}^{n}$.