| |
|
|
|
|
|
|
|
|
1. |
Record Nr. |
UNINA9910973246903321 |
|
|
Autore |
Flenner H |
|
|
Titolo |
Cancellation for Surfaces Revisited |
|
|
|
|
|
Pubbl/distr/stampa |
|
|
Providence : , : American Mathematical Society, , 2022 |
|
©2022 |
|
|
|
|
|
|
|
|
|
ISBN |
|
|
|
|
|
|
|
|
Edizione |
[1st ed.] |
|
|
|
|
|
Descrizione fisica |
|
1 online resource (124 pages) |
|
|
|
|
|
|
Collana |
|
Memoirs of the American Mathematical Society ; ; v.278 |
|
|
|
|
|
|
Classificazione |
|
|
|
|
|
|
Altri autori (Persone) |
|
|
|
|
|
|
|
|
Disciplina |
|
|
|
|
|
|
|
|
Soggetti |
|
Surfaces, Algebraic |
Cancellation theory (Group theory) |
Moduli theory |
Algebraic geometry -- Affine geometry -- Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) |
Algebraic geometry -- Families, fibrations -- Fine and coarse moduli spaces |
|
|
|
|
|
|
|
|
Lingua di pubblicazione |
|
|
|
|
|
|
Formato |
Materiale a stampa |
|
|
|
|
|
Livello bibliografico |
Monografia |
|
|
|
|
|
Nota di contenuto |
|
Cover -- Title page -- Introduction -- Chapter 1. Generalities -- 1.1. Cancellation and the Makar-Limanov invariant -- 1.2. Non-cancellation and Gizatullin surfaces -- 1.3. The Danielewski-Fieseler construction -- 1.4. Affine modifications -- Chapter 2. ¹-fibered surfaces via affine modifications -- 2.1. Covering trick and GDF surfaces -- 2.2. Pseudominimal completion and extended divisor -- 2.3. Blowup construction -- 2.4. GDF surfaces via affine modifications -- Chapter 3. Vector fields and natural coordinates -- 3.1. Locally nilpotent vertical vector fields -- 3.2. Standard affine charts -- 3.3. Natural coordinates -- 3.4. Special _{ }-quasi-invariants -- 3.5. Examples of GDF surfaces of Danielewski type -- Chapter 4. Relative flexibility -- 4.1. Definitions and the main theorem -- 4.2. Transitive group actions on Veronese cones -- 4.3. Relatively transitive group actions on cylinders -- 4.4. A relative Abhyankar-Moh-Suzuki Theorem -- Chapter 5. Rigidity of |
|
|
|
|
|
|
|
|
|
|
|
cylinders upon deformation of surfaces -- 5.1. Equivariant Asanuma modification -- 5.2. Rigidity of cylinders under deformations of GDF surfaces -- 5.3. Rigidity of cylinders under deformations of ¹-fibered surfaces -- 5.4. Rigidity of line bundles over affine surfaces -- Chapter 6. Basic examples of Zariski factors -- 6.1. Line bundles over affine curves -- 6.2. Parabolic _{ }-surfaces: an overview -- 6.3. Parabolic _{ }-surfaces as Zariski factors -- Chapter 7. Zariski 1-factors -- 7.1. Stretching and rigidity of cylinders -- 7.2. Non-cancellation for GDF surfaces -- 7.3. Extended graphs of Gizatullin surfaces -- 7.4. Zariski 1-factors and affine ¹-fibered surfaces -- Chapter 8. Classical examples -- Chapter 9. GDF surfaces with isomorphic cylinders -- 9.1. Preliminaries -- 9.2. Classification of GDF cylinders up to -isomorphism -- 9.3. GDF surfaces whose fiber trees are bushes. |
9.4. Spring bushes versus bushes -- 9.5. Cylinders over Danielewski-Fieseler surfaces -- 9.6. Proof of the main theorem -- Chapter 10. On moduli spaces of GDF surfaces -- 10.1. Coarse moduli spaces of GDF surfaces -- 10.2. The automorphism group of a GDF surface -- 10.3. Configuration spaces and configuration invariants -- 10.4. Versal deformation families of trivializing sequences -- 10.5. Proof of Theorem 10.1.3 -- Acknowledgments -- Bibliography -- Back Cover. |
|
|
|
|
|
|
Sommario/riassunto |
|
"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"-- |
|
|
|
|
|
|
|
| |