- Hacettepe Journal of Mathematics and Statistics
- Cilt: 53 Sayı: 1
- Graphs of schemes associated to group actions
Graphs of schemes associated to group actions
Authors : Ali Özgür Kişisel, Engin Özkan
Pages : 145-154
Doi:10.15672/hujms.1206439
View : 194 | Download : 344
Publication Date : 2024-02-29
Article Type : Research
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