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Cancellation for surfaces revisited
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 admi...
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| Auteurs principaux : | , , |
|---|---|
| Format : | Livre |
| Langue : | anglais |
| Titre complet : | Cancellation for surfaces revisited / H. Flenner, S. Kaliman, M. Zaidenberg |
| Publié : |
Providence (R.I.) :
American Mathematical Society
, 2022 |
| Description matérielle : | 1 vol. (V-111 p.) |
| Collection : | Memoirs of the American Mathematical Society ; 1371 |
| Sujets : | |
| Documents associés : | Autre format:
Cancellation for surfaces revisited |
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| 200 | 1 | |a Cancellation for surfaces revisited |f H. Flenner, S. Kaliman, M. Zaidenberg | |
| 214 | 0 | |a Providence (R.I.) |c American Mathematical Society |d 2022 | |
| 215 | |a 1 vol. (V-111 p.) |d 26 cm | ||
| 225 | 2 | |a Memoirs of the American Mathematical Society |x 0065-9266 |v number 1371 | |
| 320 | |a Bibliogr. p. 109-111 | ||
| 330 | |a 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. |2 résumé des auteurs | ||
| 359 | 2 | |b Generalities |b A -fibered surfaces via affine modifications |b Vector fields and natural coordinates |b Relative flexibility |b Rigidity of cylinders upon deformation of surfaces |b Basic examples of Zariski factors |b Zariski 1-factors |b Classical examples |b GDF surfaces with isomorphic cylinders |b On moduli spaces of GDF surfaces | |
| 410 | | | |0 013293931 |t Memoirs of the American Mathematical Society |x 0065-9266 |v 1371 | |
| 452 | | | |t Cancellation for surfaces revisited |y 978-1-4704-7171-2 | |
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