Graphs of schemes associated to group actions

Year 2024, Volume: 53 Issue: 1, 145 - 154, 29.02.2024

Ali Özgür Kişisel , Engin Özkan

https://doi.org/10.15672/hujms.1206439

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 proof of Hartshorne's theorem on the connectedness of the Hilbert scheme in the case of $d$ points in $\mathbb{P}^{n}$.
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Keywords

Hilbert scheme, Borel group action, A-graph

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