Cancellation for surfaces revisited / H. Flenner, S. Kaliman, M. Zaidenberg.
| Author/creator | Flenner, H. |
| Other author | Kaliman, S. (Shulim |
| Other author | Zaidenberg, Mikhail |
| Format | Electronic |
| Publication Info | Providence, RI : AMS, American Mathematical Society, 2022. |
| Description | v, 111 pages 24 cm |
| Supplemental Content | Full text available from Ebook Central - Academic Complete |
| Subjects |
| Series | Memoirs of the American Mathematical Society, 0065-9266 ; volume 278, number 1371 |
| Contents | Generalities -- A℗£-fibered surfaces via affine modifications -- Vector fields and natural coordinates -- Relative flexibility -- Rigidity of cylinders upon deformation of surfaces -- Basic examples of Zariski factors -- Zariski 1-factors -- Classical examples -- GDF surfaces with isomorphic cylinders -- On moduli spaces of GDF surfaces. |
| Abstract | "The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism X An X An for (affine) algebraic varieties X and X implies that X X. In this paper we provide a criterion for cancellation by the affine line (that is, n 1) in the case where X is a normal affine surface admitting an A1-fibration X B with no multiple fiber over a smooth affine curve B. For two such surfaces X B and X B we give a criterion as to when the cylinders X A1 and X A1 are isomorphic over B. The latter criterion is expressed in terms of linear equivalence of certain divisors on the Danielewski-Fieseler quotient of X over B. It occurs that for a smooth A1-fibered surface X B the cancellation by the affine line holds if and only if X B is a line bundle, and, for a normal such X, if and only if X B is a cyclic quotient of a line bundle (an orbifold line bundle). If X does not admit any A1-fibration over an affine base then the cancellation by the affine line is known to hold for X by a result of Bandman and Makar-Limanov. If the cancellation does not hold then X deforms in a non-isotrivial family of A1-fibered surfaces B with cylinders A1 isomorphic over B. We construct such versal deformation families and their coarse moduli spaces provided B does not admit nonconstant invertible functions. Each of these coarse moduli spaces has infinite number of irreducible components of growing dimensions; each component is an affine variety with quotient singularities. Finally, we analyze from our viewpoint the examples of non-cancellation constructed by Danielewski, tom Dieck, Wilkens, Masuda and Miyanishi, e.a"-- Provided by publisher. |
| Bibliography note | Includes bibliographical references. |
| Access restriction | Available only to authorized users. |
| Technical details | Mode of access: World Wide Web |
| Genre/form | Electronic books. |
| LCCN | 2022027943 |
| ISBN | 9781470453732 paperback |
| ISBN |