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 $\mathbb{P}^{n}$.
.
Primary Language | English |
---|---|
Subjects | Mathematical Sciences |
Journal Section | Mathematics |
Authors | |
Early Pub Date | January 10, 2024 |
Publication Date | February 29, 2024 |
Published in Issue | Year 2024 Volume: 53 Issue: 1 |