Graphs of schemes associated to group actions

Kişisel A. U. Ö., Özkan E.

Hacettepe Journal of Mathematics and Statistics, vol.53, no.1, pp.145-154, 2024 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 53 Issue: 1
  • Publication Date: 2024
  • Doi Number: 10.15672/hujms.1206439
  • Journal Name: Hacettepe Journal of Mathematics and Statistics
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, zbMATH
  • Page Numbers: pp.145-154
  • Keywords: A-graph, Borel group action, Hilbert scheme
  • Middle East Technical University Affiliated: Yes

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 ℙn.