$\mathbb{A}^1$-cylinders over smooth affine surfaces of negative Kodaira dimension
Résumé
The Zariski Cancellation problem for smooth affine surfaces asks whether two suchsurfaces whose products with the affine line are isomorphic are isomorphic themselves. Byresults of Iitaka-Fujita, the answer is positive for surfaces of non-negative Kodaira dimen-sion. By a characterization due to Miyanishi, surfaces of negative Kodaira dimension arefibered by the affine line, and by a celebrated result of Miyanishi-Sugie, the answer to theproblem is positive if one of the surfaces is the affine plane. On the other hand, exam-ples of non-isomorphicA1-fibered affine surfaces with isomorphicA1-cylinders were firstconstructed by Danielewski in 1989, and then by many other authors. All these counter-examples are essentially constructed by variants of the method employed by Danielewski,nowadays known as the ”Danielewski fiber product trick”. In this talk, I will explain thatthis method is actually more than a trick: re-interpreted in a suitable way, it providesa necessary and sufficient criterion for twoA1-fibered surfaces over a same affine basecurveCto have isomorphic relativeA1-cylinders overC. The characterization can beroughly stated as follows: two such surfaces have isomorphic relativeA1-cylinders overCif and only if their respectively relative (log)-canonical classes are equal when viewed ascertain naturally definedQ-divisors on a non-separated orbifold curve ̆CdominatingC,fully determined by the structure of the fibers of theA1-fibrations at hand. (Joint workin progress with S. Kaliman and M. Zaidenberg).