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Cancellation for Surfaces Revisited
About this Title
H. Flenner, S. Kaliman and M. Zaidenberg
Publication: Memoirs of the American Mathematical Society
Publication Year:
2022; Volume 278, Number 1371
ISBNs: 978-1-4704-5373-2 (print); 978-1-4704-7171-2 (online)
DOI: https://doi.org/10.1090/memo/1371
Published electronically: June 6, 2022
Keywords: Cancellation,
affine surface,
group action,
one-parameter subgroup,
transitivity
Table of Contents
Chapters
- Introduction
- 1. Generalities
- 2. $\mathbb {A}^1$-fibered surfaces via affine modifications
- 3. Vector fields and natural coordinates
- 4. Relative flexibility
- 5. Rigidity of cylinders upon deformation of surfaces
- 6. Basic examples of Zariski factors
- 7. Zariski 1-factors
- 8. Classical examples
- 9. GDF surfaces with isomorphic cylinders
- 10. On moduli spaces of GDF surfaces
Abstract
The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism $X\times \mathbb {A}^n\cong X’\times \mathbb {A}^n$ for (affine) algebraic varieties $X$ and $X’$ implies that $X\cong X’$. 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 $\mathbb {A}^1$-fibration $X\to B$ with no multiple fiber over a smooth affine curve $B$. For two such surfaces $X\to B$ and $X’\to B$ we give a criterion as to when the cylinders $X\times \mathbb {A}^1$ and $X’\times \mathbb {A}^1$ 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 $\mathbb {A}^1$-fibered surface $X\to B$ the cancellation by the affine line holds if and only if $X\to B$ is a line bundle, and, for a normal such $X$, if and only if $X\to B$ is a cyclic quotient of a line bundle (an orbifold line bundle). If $X$ does not admit any $\mathbb {A}^1$-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 $\mathbb {A}^1$-fibered surfaces $X_\lambda \to B$ with cylinders $X_\lambda \times \mathbb {A}^1$ 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.
- Shreeram S. Abhyankar, William Heinzer, and Paul Eakin, On the uniqueness of the coefficient ring in a polynomial ring, J. Algebra 23 (1972), 310–342. MR 306173, DOI 10.1016/0021-8693(72)90134-2
- I. Arzhantsev, H. Flenner, S. Kaliman, F. Kutzschebauch, and M. Zaidenberg, Flexible varieties and automorphism groups, Duke Math. J. 162 (2013), no. 4, 767–823. MR 3039680, DOI 10.1215/00127094-2080132
- Ivan Arzhantsev, Ulrich Derenthal, Jürgen Hausen, and Antonio Laface, Cox rings, Cambridge Studies in Advanced Mathematics, vol. 144, Cambridge University Press, Cambridge, 2015. MR 3307753
- I. V. Arzhantsev and S. A. Gaĭfullin, Cox rings, semigroups, and automorphisms of affine varieties, Mat. Sb. 201 (2010), no. 1, 3–24 (Russian, with Russian summary); English transl., Sb. Math. 201 (2010), no. 1-2, 1–21. MR 2641086, DOI 10.1070/SM2010v201n01ABEH004063
- I. V. Arzhantsev, M. G. Zaĭdenberg, and K. G. Kuyumzhiyan, Flag varieties, toric varieties, and suspensions: three examples of infinite transitivity, Mat. Sb. 203 (2012), no. 7, 3–30 (Russian, with Russian summary); English transl., Sb. Math. 203 (2012), no. 7-8, 923–949. MR 2986429, DOI 10.1070/SM2012v203n07ABEH004248
- Teruo Asanuma, Non-linearizable algebraic $k^\ast$-actions on affine spaces, Invent. Math. 138 (1999), no. 2, 281–306. MR 1720185, DOI 10.1007/s002220050379
- Tatiana M. Bandman and Leonid Makar-Limanov, Cylinders over affine surfaces, Japan. J. Math. (N.S.) 26 (2000), no. 1, 207–217. MR 1771430, DOI 10.4099/math1924.26.207
- T. Bandman and L. Makar-Limanov, Affine surfaces with $\textrm {AK}(S)=\Bbb C$, Michigan Math. J. 49 (2001), no. 3, 567–582. MR 1872757, DOI 10.1307/mmj/1012409971
- Tatiana Bandman and Leonid Makar-Limanov, Nonstability of the AK invariant, Michigan Math. J. 53 (2005), no. 2, 263–281. MR 2152699, DOI 10.1307/mmj/1123090767
- T. Bandman, L. Makar-Limanov. Affine surfaces with isomorphic cylinders. Unpublished notes, Bar Ilan University 2006, 17p.
- T. Bandman, L. Makar-Limanov. Non-stability of AK-invariant for some $\mathbb {Q}$-planes. Unpublished notes, Bar Ilan University 2006, 8p.
- A. Crachiola and L. Makar-Limanov, On the rigidity of small domains, J. Algebra 284 (2005), no. 1, 1–12. MR 2115001, DOI 10.1016/j.jalgebra.2004.09.015
- Anthony J. Crachiola and Leonid G. Makar-Limanov, An algebraic proof of a cancellation theorem for surfaces, J. Algebra 320 (2008), no. 8, 3113–3119. MR 2450715, DOI 10.1016/j.jalgebra.2008.03.037
- Anthony J. Crachiola and Stefan Maubach, Rigid rings and Makar-Limanov techniques, Comm. Algebra 41 (2013), no. 11, 4248–4266. MR 3169516, DOI 10.1080/00927872.2012.695832
- David A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), no. 1, 17–50. MR 1299003
- Daniel Daigle, Locally nilpotent derivations and Danielewski surfaces, Osaka J. Math. 41 (2004), no. 1, 37–80. MR 2040063
- W. Danielewski. On a cancellation problem and automorphism groups of affine algebraic varieties. Preprint, Warsaw, 1989.
- Robert Dryło, Non-uniruledness and the cancellation problem. II, Ann. Polon. Math. 92 (2007), no. 1, 41–48. MR 2318509, DOI 10.4064/ap92-1-4
- Adrien Dubouloz, Danielewski-Fieseler surfaces, Transform. Groups 10 (2005), no. 2, 139–162. MR 2195597, DOI 10.1007/s00031-005-1004-x
- A. Dubouloz, Embeddings of Danielewski surfaces in affine spaces, Comment. Math. Helv. 81 (2006), no. 1, 49–73. MR 2208797, DOI 10.4171/CMH/42
- A. Dubouloz. Quelques remarques sur la notion de modification affine. arXiv:math/0503142 (2005), 5p.
- Adrien Dubouloz, The cylinder over the Koras-Russell cubic threefold has a trivial Makar-Limanov invariant, Transform. Groups 14 (2009), no. 3, 531–539. MR 2534798, DOI 10.1007/s00031-009-9051-3
- Adrien Dubouloz, Flexible bundles over rigid affine surfaces, Comment. Math. Helv. 90 (2015), no. 1, 121–137. MR 3317335, DOI 10.4171/CMH/348
- Adrien Dubouloz, Affine surfaces with isomorphic $\Bbb A^2$-cylinders, Kyoto J. Math. 59 (2019), no. 1, 181–193. MR 3934627, DOI 10.1215/21562261-2018-0005
- Adrien Dubouloz and Pierre-Marie Poloni, On a class of Danielewski surfaces in affine 3-space, J. Algebra 321 (2009), no. 7, 1797–1812. MR 2494748, DOI 10.1016/j.jalgebra.2008.12.009
- Karl-Heinz Fieseler, On complex affine surfaces with $\textbf {C}^+$-action, Comment. Math. Helv. 69 (1994), no. 1, 5–27. MR 1259603, DOI 10.1007/BF02564471
- David R. Finston and Stefan Maubach, The automorphism group of certain factorial threefolds and a cancellation problem, Israel J. Math. 163 (2008), 369–381. MR 2391136, DOI 10.1007/s11856-008-0016-3
- Hubert Flenner and Mikhail Zaidenberg, Normal affine surfaces with $\Bbb C^\ast$-actions, Osaka J. Math. 40 (2003), no. 4, 981–1009. MR 2020670
- Hubert Flenner and Mikhail Zaidenberg, Locally nilpotent derivations on affine surfaces with a $\Bbb C^*$-action, Osaka J. Math. 42 (2005), no. 4, 931–974. MR 2196000
- Hubert Flenner, Shulim Kaliman, and Mikhail Zaidenberg, Completions of $\Bbb C^*$-surfaces, Affine algebraic geometry, Osaka Univ. Press, Osaka, 2007, pp. 149–201. MR 2327238
- Hubert Flenner, Shulim Kaliman, and Mikhail Zaidenberg, Uniqueness of $\Bbb C^*$- and $\Bbb C_+$-actions on Gizatullin surfaces, Transform. Groups 13 (2008), no. 2, 305–354. MR 2426134, DOI 10.1007/s00031-008-9014-0
- Hubert Flenner, Shulim Kaliman, and Mikhail Zaidenberg, Smooth affine surfaces with non-unique $\Bbb C^\ast$-actions, J. Algebraic Geom. 20 (2011), no. 2, 329–398. MR 2762994, DOI 10.1090/S1056-3911-2010-00533-4
- H. Flenner, S. Kaliman, and M. Zaidenberg. Deformation equivalence of affine ruled surfaces. arXiv:1305.5366v1 (2013), 34p.
- Robert M. Fossum, The divisor class group of a Krull domain, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 74, Springer-Verlag, New York-Heidelberg, 1973. MR 0382254
- Takao Fujita, On Zariski problem, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 3, 106–110. MR 531454
- Takao Fujita, On the topology of noncomplete algebraic surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 3, 503–566. MR 687591
- Jean-Philippe Furter and Stéphane Lamy, Normal subgroup generated by a plane polynomial automorphism, Transform. Groups 15 (2010), no. 3, 577–610. MR 2718938, DOI 10.1007/s00031-010-9095-4
- Mikio Furushima, Finite groups of polynomial automorphisms in $\textbf {C}^{n}$, Tohoku Math. J. (2) 35 (1983), no. 3, 415–424. MR 711357, DOI 10.2748/tmj/1178229000
- M. H. Gizatullin, Quasihomogeneous affine surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1047–1071 (Russian). MR 0286791
- Alexander Grothendieck, Fondements de la géométrie algébrique. [Extraits du Séminaire Bourbaki, 1957–1962.], Secrétariat mathématique, Paris, 1962 (French). MR 0146040
- R. V. Gurjar, K. Masuda, M. Miyanishi, and P. Russell, Affine lines on affine surfaces and the Makar-Limanov invariant, Canad. J. Math. 60 (2008), no. 1, 109–139. MR 2381169, DOI 10.4153/CJM-2008-005-8
- R. V. Gurjar and M. Miyanishi, Automorphisms of affine surfaces with $\Bbb A^1$-fibrations, Michigan Math. J. 53 (2005), no. 1, 33–55. MR 2125532, DOI 10.1307/mmj/1114021083
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- Shigeru Iitaka and Takao Fujita, Cancellation theorem for algebraic varieties, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 1, 123–127. MR 450278
- Shulim Kaliman, Actions of $\Bbb C^*$ and $\Bbb C_+$ on affine algebraic varieties, Algebraic geometry—Seattle 2005. Part 2, Proc. Sympos. Pure Math., vol. 80, Amer. Math. Soc., Providence, RI, 2009, pp. 629–654. MR 2483949, DOI 10.1090/pspum/080.2/2483949
- Sh. Kaliman and F. Kutzschebauch, On algebraic volume density property, Transform. Groups 21 (2016), no. 2, 451–478. MR 3492044, DOI 10.1007/s00031-015-9360-7
- Sh. Kaliman and M. Zaidenberg, Affine modifications and affine hypersurfaces with a very transitive automorphism group, Transform. Groups 4 (1999), no. 1, 53–95. MR 1669174, DOI 10.1007/BF01236662
- Sergei Kovalenko, Alexander Perepechko, and Mikhail Zaidenberg, On automorphism groups of affine surfaces, Algebraic varieties and automorphism groups, Adv. Stud. Pure Math., vol. 75, Math. Soc. Japan, Tokyo, 2017, pp. 207–286. MR 3793368, DOI 10.2969/aspm/07510207
- Herbert Lange, On elementary transformations of ruled surfaces, J. Reine Angew. Math. 346 (1984), 32–35. MR 727394, DOI 10.1515/crll.1984.346.32
- V. Lin, M. Zaidenberg. Automorphism groups of configuration spaces and discriminant varieties. arXiv:1505.06927 (2015), 61p.
- Joseph Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279. MR 276239
- L. Makar-Limanov, On the group of automorphisms of a surface $x^ny=P(z)$, Israel J. Math. 121 (2001), 113–123. MR 1818396, DOI 10.1007/BF02802499
- Kayo Masuda, Families of hypersurfaces with noncancellation property, Proc. Amer. Math. Soc. 145 (2017), no. 4, 1439–1452. MR 3601537, DOI 10.1090/proc/13489
- Kayo Masuda and Masayoshi Miyanishi, Affine pseudo-planes and cancellation problem, Trans. Amer. Math. Soc. 357 (2005), no. 12, 4867–4883. MR 2165391, DOI 10.1090/S0002-9947-05-04046-8
- Kayo Masuda and Masayoshi Miyanishi, Equivariant cancellation for algebraic varieties, Affine algebraic geometry, Contemp. Math., vol. 369, Amer. Math. Soc., Providence, RI, 2005, pp. 183–195. MR 2126662, DOI 10.1090/conm/369/06812
- Masayoshi Miyanishi, Open algebraic surfaces, CRM Monograph Series, vol. 12, American Mathematical Society, Providence, RI, 2001. MR 1800276, DOI 10.1090/crmm/012
- Masayoshi Miyanishi and Tohru Sugie, Affine surfaces containing cylinderlike open sets, J. Math. Kyoto Univ. 20 (1980), no. 1, 11–42. MR 564667, DOI 10.1215/kjm/1250522319
- Lucy Moser-Jauslin and Pierre-Marie Poloni, Embeddings of a family of Danielewski hypersurfaces and certain $\mathbf C^+$-actions on $\mathbf C^3$, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 5, 1567–1581 (English, with English and French summaries). MR 2273864
- D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906, DOI 10.1007/978-3-642-57916-5
- Masayoshi Nagata and Masaki Maruyama, Note on the structure of a ruled surface, J. Reine Angew. Math. 239(240) (1969), 68–73. MR 257085, DOI 10.1515/crll.1969.239-240.68
- P.-M. Poloni, Classification(s) of Danielewski hypersurfaces, Transform. Groups 16 (2011), no. 2, 579–597. MR 2806502, DOI 10.1007/s00031-011-9146-5
- V. L. Popov. Open Problems. In: Affine algebraic geometry, 12–16, Contemp. Math. 369, Amer. Math. Soc., Providence, RI, 2005.
- C. P. Ramanujam, A note on automorphism groups of algebraic varieties, Math. Ann. 156 (1964), 25–33. MR 166198, DOI 10.1007/BF01359978
- Walter Rudin, Preservation of level sets by automorphisms of $\textbf {C}^n$, Indag. Math. (N.S.) 4 (1993), no. 4, 489–497. MR 1252994, DOI 10.1016/0019-3577(93)90020-Y
- J.-P. Serre. Espaces fibrés algébriques. Séminaire C. Chevalley, Anneaux de Chow, Exposé 1, 1958.
- J.-P. Serre. Sur les modules projectifs. Sém. Dubreil-Pisot 14 (1960 – 61), 1–16.
- Hideyasu Sumihiro, Equivariant completion, J. Math. Kyoto Univ. 14 (1974), 1–28. MR 337963, DOI 10.1215/kjm/1250523277
- Tammo tom Dieck, Homology planes without cancellation property, Arch. Math. (Basel) 59 (1992), no. 2, 105–114. MR 1170634, DOI 10.1007/BF01190674
- Jörn Wilkens, On the cancellation problem for surfaces, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 9, 1111–1116 (English, with English and French summaries). MR 1647227, DOI 10.1016/S0764-4442(98)80071-2
- David Wright, Polynomial automorphism groups, Polynomial automorphisms and related topics, Publishing House for Science and Technology, Hanoi, 2007, pp. 1–19. MR 2389269
- M. Zaidenberg. Lecture course “Affine surfaces and the Zariski Cancellation Problem” (a program). http://www.mat.uniroma2.it/~flamini/workshops/LectZaidenberg.html
- Oscar Zariski, The reduction of the singularities of an algebraic surface, Ann. of Math. (2) 40 (1939), 639–689. MR 159, DOI 10.2307/1968949