Affine-ruled varieties without the Laurent cancellation property

TL;DR

This paper constructs specific hypersurfaces in complex affine space that serve as counterexamples to the Laurent Cancellation Problem, showing non-isomorphic varieties with isomorphic cylinders and squares.

Contribution

It introduces a new method to build hypersurfaces with isomorphic *-cylinders, providing explicit counterexamples to the Laurent Cancellation Property.

Findings

Identified hypersurfaces with isomorphic *-cylinders that are non-isomorphic

Constructed pairs of non-isomorphic varieties with isomorphic Cartesian squares

Expanded understanding of affine *-cylinders in algebraic geometry

Abstract

We describe a method to construct hypersurfaces of the complex affine $n$-space with isomorphic $\mathbb{C}^*$-cylinders. Among these hypersurfaces, we find new explicit counterexamples to the Laurent Cancellation Problem, i.e. hypersurfaces that are non isomorphic, although their $\mathbb{C}^*$-cylinders are isomorphic as abstract algebraic varieties. We also provide examples of non isomorphic varieties $X$ and $Y$ with isomorphic cartesian squares $X\times X$ and $Y\times Y$.