Stratifications of schemes using tangent vector fields

TL;DR

This paper develops a method to stratify schemes over a field of characteristic zero using tangent vector fields, creating locally closed subsets where the vector fields act transitively.

Contribution

It introduces a new stratification technique for schemes based on Lie algebra submodules of the tangent sheaf, extending the understanding of tangent vector field actions.

Findings

Constructed stratifications preserved by Lie algebra actions

Provided a framework for analyzing tangent vector fields on schemes

Enhanced the geometric understanding of scheme structures

Abstract

Let $X/k$ be a noetherian scheme over a field $k$ of characteristic 0, such that the residue field at its closed points are algebraic extensions of $k$. Let ${\mathfrak g}_{X/k}\subset T_{{X/k}}$ be an ${\mathcal O}_{X}$-submodule of the tangent sheaf which is closed under the Lie bracket. We construct a stratification of the subset of closed points in $X$ by locally closed subsets that are preserved by ${\mathfrak g}_{X/k}$, on which ${\mathfrak g}_{X/k}$ acts transitively.