LEADER 06163nam 22006133 450 001 9910973246903321 005 20231110220645.0 010 $a9781470471712 010 $a147047171X 035 $a(MiAaPQ)EBC29378995 035 $a(Au-PeEL)EBL29378995 035 $a(CKB)24267882200041 035 $a(OCoLC)1336954489 035 $a(EXLCZ)9924267882200041 100 $a20220721d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aCancellation for Surfaces Revisited 205 $a1st ed. 210 1$aProvidence :$cAmerican Mathematical Society,$d2022. 210 4$d©2022. 215 $a1 online resource (124 pages) 225 1 $aMemoirs of the American Mathematical Society ;$vv.278 311 08$aPrint version: Flenner, H. Cancellation for Surfaces Revisited Providence : American Mathematical Society,c2022 9781470453732 327 $aCover -- 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. 327 $a9.4. 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. 330 $a"The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism X An X An for (affine) algebraic varieties X and X implies that X X. In this paper we provide a criterion for cancellation by the affine line (that is, n 1) in the case where X is a normal affine surface admitting an A1-fibration X B with no multiple fiber over a smooth affine curve B. For two such surfaces X B and X B we give a criterion as to when the cylinders X A1 and X A1 are isomorphic over B. The latter criterion is expressed in terms of linear equivalence of certain divisors on the Danielewski-Fieseler quotient of X over B. It occurs that for a smooth A1-fibered surface X B the cancellation by the affine line holds if and only if X B is a line bundle, and, for a normal such X, if and only if X B is a cyclic quotient of a line bundle (an orbifold line bundle). If X does not admit any A1-fibration over an affine base then the cancellation by the affine line is known to hold for X by a result of Bandman and Makar-Limanov. If the cancellation does not hold then X deforms in a non-isotrivial family of A1-fibered surfaces B with cylinders A1 isomorphic over B. We construct such versal deformation families and their coarse moduli spaces provided B does not admit nonconstant invertible functions. Each of these coarse moduli spaces has infinite number of irreducible components of growing dimensions; each component is an affine variety with quotient singularities. Finally, we analyze from our viewpoint the examples of non-cancellation constructed by Danielewski, tom Dieck, Wilkens, Masuda and Miyanishi, e.a"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society 606 $aSurfaces, Algebraic 606 $aCancellation theory (Group theory) 606 $aModuli theory 606 $aAlgebraic geometry -- Affine geometry -- Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)$2msc 606 $aAlgebraic geometry -- Families, fibrations -- Fine and coarse moduli spaces$2msc 615 0$aSurfaces, Algebraic. 615 0$aCancellation theory (Group theory) 615 0$aModuli theory. 615 7$aAlgebraic geometry -- Affine geometry -- Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem). 615 7$aAlgebraic geometry -- Families, fibrations -- Fine and coarse moduli spaces. 676 $a516.3/52 676 $a516.352 686 $a14R10$a14D22$2msc 700 $aFlenner$b H$01800248 701 $aKaliman$b S$01800249 701 $aZaidenberg$b M$01800250 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910973246903321 996 $aCancellation for Surfaces Revisited$94344957 997 $aUNINA