Cancellation for Surfaces Revisited

The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism If the cancellation does not hold then

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Hlavní autori: Flenner, H., Kaliman, S., Zaidenberg, M.
Médium: E-kniha Kniha
Jazyk:English
Vydavateľské údaje: Providence, Rhode Island American Mathematical Society 2022
Vydanie:1
Edícia:Memoirs of the American Mathematical Society
Predmet:
Algebraic geometry > Affine geometry > Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem). msc
Algebraic geometry > Families, fibrations > Fine and coarse moduli spaces. msc
Cancellation theory (Group theory)
Moduli theory
Surfaces, Algebraic
Cancellation
affine surface
group action
transitivity
one-parameter subgroup
ISBN:1470453738, 9781470453732
ISSN:0065-9266, 1947-6221
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Obsah:
  • Introduction -- Generalities -- <inline-formula content-type="math/mathml"> A 1 \mathbb {A}^1 </inline-formula>-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
  • 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