Vénéreau polynomials and related fiber bundles
by
Mikhail Zaidenberg
Institut Fourier, Grenoble
Coauthors: Shulim Kaliman (Miami)

The Vénéreau polynomials on A⁴=A_C⁴,

vn:=y+xn(xz+y(yu+z2)),        n ³ 1,

have all the fibers isomorphic to the affine space A³. Moreover, for all n ³ 1 the map (v_n, x):A⁴® A² yields a flat family of affine planes over A². It occurs that, over the punctured plane A²\{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].

This is an open question whether v₁ and v₂ are variables of the polynomial ring C^[4]=C[x, y, z, u], whereas S. Vénéreau established that  v_n is indeed a variable of C[x][y, z, u] over C[x] for n ³ 3. We will discuss another proof of this Vénéreau's result based on the above equivalence, 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.

Paper reference: arXiv:math.AG/0308191

Date received: January 20, 2005