Vénéreau polynomials and related fiber bundles

The Vénéreau polynomials v_n:y+xⁿ(xz+y(yu+z²)), n≥1, on A_ℂ ⁴ have all fibers isomorphic to the affine space A_ℂ³. Moreover, for all n≥1 the map (v_n,x):A_ℂ⁴→ A_ℂ² yields a flat family of affine planes over A_ℂ². In the present note we show that over the punctured plane A_ℂ²\{0̄}, this family is a fiber bundle. This bundle is trivial if and only if v_n is a variable of the ring ℂ[x][y,z,u] over ℂ[x]. It is an open question whether v₁ and v₂ are variables of the polynomial ring ℂ^[4]=ℂ[x,y,z,u], whereas Vénéreau established that v_n is indeed a variable of ℂ[x][y,z,u] over ℂ[x] for n≥3. In this note we give another proof of Vénéreau's result based on the above equivalence. We also discuss some other equivalent properties, as well as the relations to the Abhyankar-Sathaye Embedding Problem and to the Dolgachev-Weisfeiler Conjecture on triviality of flat families with fibers affine spaces.