Vénéreau polynomials and related fiber bundles

AbstractThe Vénéreau polynomialsvn≔y+xn(xz+y(yu+z2)),n⩾1,on AC4 have all fibers isomorphic to the affine space AC3. Moreover, for all n⩾1 the map (vn,x):AC4→AC2 yields a flat family of affine planes over AC2. In the present note we show that over the punctured plane AC2⧹{0̄}, this family is a fiber bundle. This bundle is trivial if and only if vn 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 vn 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