Auto-Arcs of Complete Intersection Varieties

TL;DR

This paper explores auto-arc spaces of complete intersection varieties, revealing their structure as flat deformations of jet schemes and introducing motivic volume concepts for curves.

Contribution

It advances the understanding of auto-arc spaces by analyzing their structure as flat deformations in complete intersections, extending previous trivial deformation results.

Findings

Auto-arc spaces of complete intersections are flat deformations of jet schemes.

The study introduces the concept of motivic volume for curves within this framework.

The work generalizes previous results on auto-arc spaces and jet schemes.

Abstract

We systematically study the so-called auto-arc spaces. Auto-arc spaces were originally introduced by Schoutens in 2012 and later generalized and studied by the author in his PhD Thesis and subsequent work. In that aforementioned work, only results concerning trivial deformations were explicitly considered because even in that case auto-arc spaces being a subset of generalized jet schemes are difficult to understand. The major advance in this work is obtained by considering auto arc spaces of complete intersections. It is shown that over $k[t]/(t^{n+1})$, these spaces can be viewed as global flat deformations over $\mathbb{A}_k^n$ of the classical jet scheme of order $n$. We propose the project in general of investigating the flat locus of this naturally induced morphism as a type of relativized version of previous results by Mustata on jet schemes a local complete intersections. We also introduce the study of so-called strong/weak deformations of curves in this context, and we show that a motivic volume can be defined in this case.