An Algebraic characterization of the affine three space in arbitrary characteristic

TL;DR

This paper provides an algebraic characterization of the affine three-space over any characteristic and explores the structure of certain algebraic varieties, contributing to the understanding of the cancellation problem in algebraic geometry.

Contribution

It introduces a new algebraic characterization of affine three-space applicable in arbitrary characteristic and applies it to analyze specific algebraic structures related to the cancellation problem.

Findings

Characterization of affine 3-space in arbitrary characteristic

Results on ML and ML* invariants for specific algebraic structures

Partial progress on the strong cancellation conjecture for k^{[2]}

Abstract

We give an algebraic characterization of the affine $3$-space over an algebraically closed field of arbitrary characteristic. We use this characterization to reformulate the following question. Let $$A=k[X, Y, Z, T]/(XY+Z^{p^e}+T+T^{sp})$$ where $p^e\nmid sp$, $sp\nmid p^e$, $e, s\geq 1$ and $k$ is an algebraically closed field of positive characteristic $p$. Is $A= k^{[3]}$? We prove some results on ML and ML$^*$ invariants and use them to prove a special case of the strong cancellation of $k^{[2]}$.