Mathematics – Algebraic Geometry
Scientific paper
2003-08-20
Mathematics
Algebraic Geometry
LaTeX, 11 pages
Scientific paper
The Venereau polynomials v-n:=y+x^n(xz+y(yu+z^2)), n>= 1, on A4 have all fibers isomorphic to the affine space A3. Moreover, for all n>= 1 the map (v-n, x) : A4 -> A2 yields a flat family of affine planes over A2. In the present note we show 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]. It is an open question whether v1 and v2 are variables of the polynomial ring C[x,y,z,u]. S. Venereau 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 an alternative proof of Venereau'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.
Kaliman Shulim
Zaidenberg Mikhail
No associations
LandOfFree
Venereau polynomials and related fiber bundles does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Venereau polynomials and related fiber bundles, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Venereau polynomials and related fiber bundles will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-167266