Venereau-type polynomials as potential counterexamples

TL;DR

This paper investigates Venereau polynomials, demonstrating their properties as hyperplanes and residual coordinates, and establishing conditions under which they are C[x]-coordinates or stably tame, thus contributing to the understanding of potential counterexamples to key algebraic conjectures.

Contribution

The paper introduces Venereau-type polynomials, proves they are hyperplanes and residual coordinates, and identifies which are C[x]-coordinates or stably tame, advancing the study of counterexamples in algebraic geometry.

Findings

b_2 is a C[x]-coordinate

Venereau-type polynomials are hyperplanes and residual coordinates

Some are proven to be C[x]-coordinates and stably tame

Abstract

We study some properties of the Venereau polynomials b_m=y+x^m(xz+y(yu+z^2)), a sequence of proposed counterexamples to the Abhyankar-Sathaye embedding conjecture and the Dolgachev-Weisfeiler conjecture. It is well known that these are hyperplanes and residual coordinates, and for m at least 3, they are C[x]-coordinates. For m=1,2, it is only known that they are 1-stable C[x]-coordinates. We show that b_2 is in fact a C[x]-coordinate. We introduce the notion of Venereau-type polynomials, and show that these are all hyperplanes, and residual coordinates. We show that some of these Venereau-type polynomials are in fact C[x]-coordinates; the rest remain potential counterexamples to the embedding and other conjectures. For those that we show to be coordinates, we also show that any automorphism with one of them as a component is stably tame. The remainder are stably tame, 1-stable C[x]-coordinates.