Greenberg approximation and the geometry of arc spaces

TL;DR

This paper explores the differential properties of generalized arc schemes and their geometric aspects, proving an approximation result for arcs by algebraic curves, and extending Kolchin's Irreducibility Theorem to arbitrary base fields.

Contribution

It introduces new insights into the geometry of arc spaces and generalizes key theorems to broader settings, including arbitrary base fields.

Findings

Proved an approximation result for arcs by algebraic curves

Studied differential properties of generalized arc schemes

Extended Kolchin's Irreducibility Theorem to arbitrary base fields

Abstract

We study the differential properties of generalized arc schemes, and geometric versions of Kolchin's Irreducibility Theorem over arbitrary base fields. As an intermediate step, we prove an approximation result for arcs by algebraic curves.