On Generalised Danielewski and Asanuma varieties

TL;DR

This paper extends the Cancellation Problem results to arbitrary characteristic fields and analyzes invariants of generalized Asanuma varieties, providing conditions for their isomorphism to polynomial rings.

Contribution

It generalizes Dubouloz's Cancellation Problem results to all characteristics and characterizes when generalized Asanuma varieties are polynomial rings.

Findings

Extended Cancellation Problem to arbitrary characteristic fields.

Described Makar-Limanov and Derksen invariants for generalized Asanuma varieties.

Provided necessary and sufficient conditions for isomorphism to polynomial rings.

Abstract

In this paper we extend a result of Dubouloz on the Cancellation Problem in higher dimensions ($\geqslant 2$) over the field of complex numbers to fields of arbitrary characteristic. We then apply the generalised result to describe the Makar-Limanov and Derksen invariant of generalised Asanuma varieties under certain hypotheses. We also establish a necessary and sufficient condition for certain generalised Asanuma varieties to be isomorphic to polynomial rings.