Graphs of schemes associated to group actions

Authors : Ali Özgür Kişisel, Engin Özkan

Pages : 145-154

Doi:10.15672/hujms.1206439

View : 215 | Download : 351

Publication Date : 2024-02-29

Article Type : Research Paper

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 $\\mathbb{P}^{n}$. .
Keywords : Hilbert scheme, Borel group action, A grap