Vénéreau polynomials and related fiber bundles

The Vénéreau polynomials v n≔y+x n(xz+y(yu+z 2)), n⩾1, on A C 4 have all fibers isomorphic to the affine space A C 3 . Moreover, for all n⩾1 the map (v n,x): A C 4→ A C 2 yields a flat family of affine planes over A C 2 . In the present note we show that over the punctured plane A C 2⧹{ 0 ̄ } , 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 v 1 and v 2 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.