Venereau polynomials and related fiber bundles
| dc.creator | Kaliman, Shulim | |
| dc.creator | Zaidenberg, Mikhail | |
| dc.date | 2003-08-20 | |
| dc.date.accessioned | 2026-07-07T05:00:31Z | |
| dc.date.available | 2026-07-07T05:00:31Z | |
| dc.description | 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. | |
| dc.description | LaTeX, 11 pages | |
| dc.identifier | https://arxiv.org/abs/math/0308191 | |
| dc.identifier | http://arxiv.org/abs/math/0308191 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/68351 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 14R10, 14R25 | |
| dc.title | Venereau polynomials and related fiber bundles | |
| dc.type | text |