Challenging problems on affine $n$-space

 Access to full text

 Full (PDF)

1. [AC95] N. A'campo, M. Oka - Geometry of plane curves via Tschirnhausen resolution towers. J. Alg. Geometry (1995) (to appear). 
2. [AbM75] S.S. Abhyankar, T.T. Moh - Embeddings of the line in the plane. J. Reine Angew. Math.276 (1975) 149-166. Zbl0332.14004MR379502
3. [As94] T. Asanuma - Non-linearizable algebaic group actions on An. J. Algebra166 (1994) 72-79. Zbl0817.14027
4. [Ba84] H. Bass - A non-triangular action of Ga on A3. J. Pure Appl. Algebra33 (1984) 1-5. Zbl0555.14019MR750225
5. [BaH85] H. Bass, W. Haboush - Linearizing certain reductive group actions. Trans. Amer. Math. Soc.292 (1985) 463-482. Zbl0602.14047MR808732
6. [BaH87] H. Bass, W. Haboush - Some equivariant K-theory of affine algebraic group actions. Comm. Algebra15 (1987) 181-217. Zbl0612.14047MR876977
7. [Bi66] A. Białynicki-Birula - Remarks on the action of an algebraic torus on kn, I. Bull. Acad. Polon. Sci. Sér. Sci. Math.14 (1966) 177-181. Zbl0163.42901
8. [Bi67] A. Białynicki-Birula - Remarks on the action of an algebraic torus on kn, II. Bull. Acad. Polon. Sci. Sér. Sci. Math.15 (1967) 123-125. Zbl0163.42901
9. [Bi76] A. Białynicki-Birula - Some properties of the decompositions of algebraic varieties determined by actions of a torus. Bull. Acad. Polon. Sci. Sér. Sci. Math.24 (1976) 667-674. Zbl0355.14015
10. [CaL70] I. Canals, E. Lluis - Acerca de un resultado de Segre. Anales del Inst. de Metam., Univ. Nac. Aut. México10 (1970) 1-15. Zbl0234.14011MR286794
11. [Da89] W. Danielewski - On the cancellation problem and automorphism group of affine algebraic varieties. Preprint 1989. (Appendix by K. Fieseler) 
12. [Di90] A. Dimca - Hypersurfaces in Cn diffeomorphic to R4n-2. Max-Planck Institut Bonn, preprint 1990. 
13. [Di92] A. Dimca - Singularities and topology of hypersurfaces. Universitext, Springer Verlag, Berlin-Heidelberg-New York, 1992. Zbl0753.57001MR1194180
14. [DMP89] K.-H. Dovermann, M. Masuda, T. Petrie - Fixed point free algebraic actions on varieties diffeomorphic to Rn. In: Topological methods in algebraic transformation groups (H. Kraft et al., eds.). Progress in Math. vol. 80, Birkhäuser Verlag, Basel - Boston, 1989, pp. 47-80. Zbl0695.57021MR1040857
15. [EaH73] P. Eakin, W. Heinzer - A cancellation problem for rings. In: Conference on Commutative Algebra (Brewer, J.W.; Rutter, E.A., eds.). Lecture Notes in Mathematics vol. 311, Springer Verlag, Berlin-Heidelberg-New York, 1973, pp. 61-77. Zbl0271.13010MR349664
16. [Fa95] M. Fankhauser - Fixed points for reductive group actions on acyclic varieties. Ann. Inst. Fourier (1995) to appear. (Thesis Basel1994) MR1370747
17. [Fi94] K.-H. Fieseler - On complex affine surfaces with C+-action. Comment. Math. Helv.69 (1994) 5-27. Zbl0806.14033MR1259603
18. [Fu79] T. Fujita - On Zariski problem. Proc. Japan Acad. Ser. A. Math. Sci.55 (1979) 106-110. Zbl0444.14026MR531454
19. [Fu81] T. Fujita - Cancellation problem for complete varieties. Invent. Math.64 (1981) 119-121. Zbl0462.14001MR621772
20. [Fu82] T. Fujita - On the topology of non-complete surfaces. J. Fac. Sci. Univ. Tokyo29 (1982) 503-566. Zbl0513.14018MR687591
21. [GiD75] M.H. Gizatullin, V.I. Danilov - Automorphisms of affine surfaces, I. Math. USSR-Izv.9 (1975) 493-534. Zbl0331.14007MR376701
22. [Gu88] J. Gubeladze - Anderson's conjecture and the maximal monoid class over which projective modules are free. Math. USSR-Sb.63 (1988) 165-180. Zbl0668.13011MR937805
23. [GuM87] R.V. Gurjar, M. Miyanishi - Affine surfaces with κ ≤ 1. In: "Algebraic Geometry and Commutative Algebra" in Honor of Masayoshi Nagata., 1987, pp. 99-124. Zbl0707.14015
24. [GuM95] R.V. Gurjar, M. Miyanishi - On contractible curves in C2. Preprint 1995. 
25. [HeK94] P. Heinzner, F. Kutzschebauch - An equivariant version of Grauert's Oka principle. Invent. Math.119 (1994) 317-346. Zbl0837.32004MR1312503
26. [Ho72] M. Hochster - Nonuniqueness of coefficient rings in a polynomial ring. Proc. Amer. Math. Soc.34 (1972) 81-82. Zbl0233.13012MR294325
27. [HsS86] W. Hsiang, E. Straume - Actions of compact connected Lie groups on acyclic manifolds with low dimensional orbit space. J. Reine Angew. Math.369 (1986) 21-39. Zbl0583.57025MR850627
28. [Ii82] S. Iitaka - Algebraic Geometry : An introduction to birational geometry of algebraic varieties. Graduate Texts in Mathematics, vol. 76, Springer Verlag, Berlin-Heidelberg-New York, 1982. Zbl0491.14006MR637060
29. [IiF77] S. Iitaka, T. Fujita - Cancellation theorem for algebraic varieties. J. Fac. Sci. Univ. Tokyo24 (1977) 123-127. Zbl0353.14013MR450278
30. [Je87] Z. Jelonek - The extension of regular and rational embeddings. Math. Ann.113 (1987) 113-120. Zbl0611.14010MR884649
31. [Je91] Z. Jelonek - Identity sets for polynomial automorphisms. J. Pure Appl. Algebra76 (1991) 333-339. Zbl0752.14010MR1147306
32. [Je92] Z. Jelonek - The Jacobian Conjecture and the extension of a polynomial embeddings. Math. Annalen294 (1992) 289-293. Zbl0757.14007MR1183407
33. [Je93] Z. Jelonek - Irreducible identity sets for polynomial automorphisms. Math. Zeitschrift212 (1993) 601-617. Zbl0806.14011MR1214049
34. [Je94] Z. Jelonek - Affine smooth varieties with finite group of automorphisms. Math. Zeitschrift216 (1994) 575-591. Zbl0824.14012MR1288046
35. [Ju42] H.W.E. Jung - Über ganze birationale Transformationen der Ebene. J. Reine Angew. Math.184 (1942) 161-174. Zbl0027.08503MR8915JFM68.0382.01
36. [Ka91] S. Kaliman - Extensions of isomorphisms between affine algebraic subvarieties of kn to automorphisms of kn. Proc. Amer. Math. Soc.113 (1991) 325-334. Zbl0743.14011MR1076575
37. [Ka93] S. Kaliman - Smooth contractible hypersurfaces in Cn and exotic algebraic structures on C3. Math. Zeitschrift214 (1993) 499-510. Zbl0792.57013MR1245209
38. [Ka95] S. Kaliman - Exotic structures on Cn and C*-actions on C3. In: Proceedings on "Complex Analysis and Geometry". Lecture Notes in Pure and Appl. Math., Marcel Dekker Inc., 1995 (to appear) Zbl0870.58007MR1365979
39. [KaM95a] S. Kaliman, L. Makar-Limanov - Affine algebraic manifolds without dominant morphisms from euclidean space. Preprint 1995. MR1466159
40. [KaM95b] S. Kaliman, L. Makar-Limanov - On Russell's contractible threefolds. Preprint 1995. 
41. [Kam75] T. Kambayashi - On the absence of nontrivial separable forms of the affine plane. J. Algebra35 (1975) 449-456. Zbl0309.14029MR369380
42. [Kam79] T. Kambayashi - Automorphism group of a polynomial ring and algebraic group action on an affine space. J. Algebra60 (1979) 439-451. Zbl0429.14017MR549939
43. [Kam80] T. Kambayashi - On Fujita's strong cancellation theorem for the affine plane. J. Fac. Sci. Univ. Tokyo, Sec. IA 27 (1980) 535-548. Zbl0453.14015MR603951
44. [KaR82] T. Kambayashi, P. Russell - On linearizing algebraic torus actions. J. Pure Appl. Algebra23 (1982) 243-250. Zbl0518.14026MR644276
45. [Kn91] F. Knop - Nichtlinearisierbare Operationen halbeinfacher Gruppen auf affinen Räumen. Invent. Math.105 (1991) 217-220. Zbl0739.20019MR1109627
46. [KoR86] M. Koras, P. Russell - Gm-actions on A3. In: Canad. Math. Soc. Confer. Proc.. vol. 6, 1986, pp. 269-276. Zbl0591.14034MR846023
47. [KoR89a] M. Koras, P. Russell - Codimension 2 torus actions on affine n-space. In: Canad. Math. Soc. Confer. Proc.. Proceedings of the Conference on "Group Actions and Invariant Theory", Montreal1988 vol. 10, 1989, pp. 103-110. Zbl0725.14036MR1021282
48. [KoR89b] M. Koras, P. Russell - On linearizing "good" C*-actions on C3. In: Canad. Math. Soc. Confer. Proc.. Proceedings of the Conference on "Group Actions and Invariant Theory" , Montreal1988 vol. 10, 1989, pp. 93-102. Zbl0687.14037MR1021281
49. [KoR95] M. Koras, P. Russell - Contractible algebraic threefolds and C*-actions on C3. Preprint 1995. 
50. [Kr89a] H. Kraft - Algebraic automorphisms of affine space. In: Topological methods in algebraic transformation groups (H. Kraft et al., eds.). Progress in Math. vol. 80, Birkhäuser Verlag, Basel-Boston, 1989, pp. 81-105. Zbl0719.14030MR1040858
51. [Kr89b] H. Kraft - G-vector bundles and the linearization problem. In: Proceedings of the Conference on "Group Actions and Invariant Theory" , Montreal1988. Can. Math. Soc. Conf. Proc. vol. 10, 1989, pp. 111-123. Zbl0703.14009MR1021283
52. [Kr90] H. Kraft - C*-actions on affine space. In: Operator Algebras, Unitary Representations, Envelopping Algebras, and Invariant Theory (A. Connes, M. Duflo, A. Joseph, R. Rentschler, eds.). Progress in Mathematics vol. 92, Birkhäuser Verlag, Basel-Boston, 1990, pp. 561-579. Zbl0747.14014MR1103604
53. [KrP85] H. Kraft, V.L. Popov - Semisimple group actions on the three dimensional affine space are linear. Comment. Math. Helv.60 (1985) 466-479. Zbl0645.14020MR814152
54. [KrS89] H. Kraft, G.W. Schwarz - Linearizing reductive group actions on affine space with one-dimensional quotient. In: Proceedings of the Conference on "Group Actions and Invariant Theory," Montreal1988. Contemp. Math. vol. 10, 1989 Zbl0704.14039MR1021284
55. [KrS92] H. Kraft, G.W. Schwarz - Reductive group actions with one-dimensional quotient. Publ. Math. IHES76 (1992) 1-97. Zbl0783.14026MR1215592
56. [KrS95] H. Kraft, G.W. Schwarz - Finite automorphisms of affine n-space. In: Proceedings of the Curaçao Conference on "Automorphisms of Affine Space". to appear, 1995 Zbl0837.14010MR1352703
57. [Li77] A. Libgober - A geometrical procedure for killing the middle dimensional homology groups of algebraic hypersurfaces. Proc. Amer. Math. Soc.63 (1977) 198-202. Zbl0359.14007MR440576
58. [Lu73] D. Luna - Slices étales. Bull. Soc. Math. France, Mémoire33 (1973) 81-105. Zbl0286.14014MR342523
59. [Ma94] L. Makar-Limanov - On the hypersurface x + x2y + z2 + t3 = 0 in C4. Preprint 1994. MR1289046
60. [MMP91] M. Masuda, L. Moser-Jauslin, T. Petrie - Equivariant algebraic vector bundles over representations of reductive groups: Applications. Proc. Natl. Acad. Sci. USA88 (1991) 9065-9066. Zbl0753.14005MR1129885
61. [MMP95] M. Masuda, L. Moser-Jauslin, T. Petrie - The equivariant Serre Problem for abelian groups. J. Amer. Math. Soc (1995). MR1380501
62. [MaP91] M. Masuda, T. Petrie - Equivariant algebraic vector bundles over representations of reductive groups: Theory. Proc. Natl. Acad. Sci. USA88 (1991) 9061-9064. Zbl0753.14042MR1129884
63. [Me95] K. Mederer - Moduli of equivariant G-vector bundles. Thesis Brandeis1995. 
64. [Mi87] M. Miyanishi - Algebraic characterization of the affine 3-space. Proc. Alg. Geometry Seminar, Singapore, World Scientific, 1987, pp. 53-67. MR966444
65. [MiS80] M. Miyanishi, T. Sugie - Affine surfaces containing cylinderlike open sets. J. Math. Kyoto Univ.20 (1980) 11-42. Zbl0445.14017MR564667
66. [MiT87] M. Miyanishi, S. Tsunoda - Open algebraic surfaces with Kodaira dimension -∞. Proc. Sympos. Pure Math.46 (1987) 435-450. Zbl0643.14021
67. [Na67] M. Nagata - A theorem on valuation rings and its applications. Nagoya Math. J.29 (1967) 85-91. Zbl0146.26302MR207688
68. [Ne89a] W.D. Neumann - On the topology of curves in complex surfaces. In: Topological methods in algebraic transformation groups (H. Kraft et al, eds.). Progress in Math. vol. 80, Birkhäuser Verlag, Basel-Boston, 1989, pp. 117-133. Zbl0715.14025MR1040860
69. [Ne89b] W.D. Neumann - Complex algebraic plane curves via their links at infinity. Invent. math.98 (1989) 445-489. Zbl0734.57011MR1022302
70. [Pa84] D.I. Panyushev - Semisimple automorphism groups of four-dimensional affine space. Math. USSR-Izv.23 (1984) 171-183. Zbl0581.14033
71. [PeR86] T. Petrie, J.D. Randall - Finite-order algebraic automorphisms of affine varieties. Comment. Math. Helv.61 (1986) 203-221. Zbl0612.14046MR856087
72. [Ra71] C.P. Ramanujam - A topological characterization of the affine plane as an algebraic variety. Ann. of Math.94 (1971) 69-88. Zbl0218.14021MR286801
73. [Re68] R. Rentschler - Opérations du groupe additif sur le plan affine. C. R. Acad. Sci. Paris267 A (1968) 384-387. Zbl0165.05402MR232770
74. [Rud82] L. Rudolph - Embedding of the line in the plane. J. Reine Angew. Math.337 (1982) 113-118. Zbl0486.14003MR676044
75. [Ru81] P. Russell - On affine-ruled rational surfaces. Math. Ann.255 (1981) 287-302. Zbl0438.14024MR615851
76. [Ru92] P. Russell - On a class of C3-like threefolds. Preliminary report 1992. 
77. [Ru95] P. Russell - Open problems on open algebraic varieties (Montreal 1994 Problems). Prépublication de l'Institut Fourier, Laboratoire Mathématiques, n° 311, Grenoble1995, 23 p.) 
78. [Sa76] A. Sathaye - On linear planes. Proc. Amer. Math. Soc.56 (1976) 1-7. Zbl0345.14013MR409472
79. [Sc89] G.W. Schwarz - Exotic algebraic group actions. C. R. Acad. Sci. Paris309 (1989) 89-94. Zbl0688.14040MR1004947
80. [Seg57] B. Segre - Corrispondenze di Möbius e transformazioni cremoniane intere. Atti Acad. Sci. Torino91 (1957) 1-17. Zbl0079.36801MR95175
81. [Se61] J.-P. Serre - Sur les modules projectifs. In Séminaire Dubreil-Pisot1960/61, n°2. Zbl0132.41301
82. [Se80] J.-P. Serre - Trees. Springer Verlag, Berlin-Heidelberg-New York, 1980. Zbl0548.20018MR607504
83. [Sh66] I.R. Shafarevich - On some infinite dimensional groups. Rend. Mat. e Appl. (5) 25 (1966) 208-212. Zbl0149.39003MR485898
84. [Sr91] V. Srinivas - On the embedding dimension of an affine variety. Math. Annalen289 (1991) 125-132. Zbl0725.14003MR1087241
85. [Su89] T. Sugie - Algebraic characterization of the affine plane and the affine 3-space. In: Topological methods in algebraic transformation groups (H. Kraft et al., eds.). Progress in Math. vol. 80, Birkhäuser Verlag, Basel - Boston, 1989, pp. 177-190. Zbl0731.14027MR1040864
86. [Suz74] M. Suzuki - Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l'espace C2. J. Math. Soc. Japan26 (1974) 241-257. Zbl0276.14001MR338423
87. [tDP89] T. Tom Dieck, T. Petrie - Homology planes: An announcement and survey. In: Topological methods in algebraic transformation groups (H. Kraft et al., eds.). Progress in Math. vol. 80, Birkhäuser Verlag, Basel - Boston, 1989, pp. 27-48. Zbl0708.14024MR1040856
88. [tDP90] T. Tom Dieck, T. Petrie - Contractible affine surfaces of Kodaira dimension one. Japan J. Math.16 (1990) 147-169. Zbl0721.14018MR1064448
89. [Ku53] W. Van Der Kulk - On polynomial rings in two variables. Nieuw Arch. Wisk.1 (1953) 33-41. Zbl0050.26002MR54574
90. [Wi90] J. Winkelmann - On free holomorphic C-actions on Cn and homogeneous Stein manifolds. Math. Ann.286 (1990) 593-612. Zbl0708.32004MR1032948
91. [Wr79] D. Wright - Abelian subgroups of Autk(k[X,Y]) and applications to actions on the affine plane. Ill. J. Math.23 (1979) 579-634. Zbl0461.20011MR540400
92. [Za91] M.G. Zaidenberg - Ramanujam surfaces and exotic algebraic structures on Cn, n ≥ 3. Soviet Math. Dokl.42 (1991) 636-640. Zbl0741.14016
93. [Za93] M.G. Zaidenberg - An analytic cancellation theorem and exotic algebraic structures on Cn, n ≥ 3. In: Astérisque217 (1993), 251-282., 1993 (Max-Planck-Institut für Mathematik, Bonn, 1991. Preprint MPI/91-26, 33p.) Zbl0801.14001
94. [Za95] M.G. Zaidenberg - On exotic algebraic structures on affine spaces. In: Proceedings of a Conference "Geometric Complex Analysis". Tokyo, March 19-29, World Scientific, 1995 (Prépublication de l'Institut Fourier, Laboratoire de Mathématiques, n° 309, Grenoble1995, 32 p.) Zbl0934.14042MR1453650
95. [ZaL83] M.G. Zaidenberg, V.Ya. Lin - An irreducible simply connected algebraic curve in C2 is a quasihomogeneous curve. Soviet Math. Dokl.28 (1983). Zbl0564.14014

1. Philippe Bonnet, Relative exactness modulo a polynomial map and algebraic $({ℂ}^p,+)$-actions
2. Sandra Marcello, Sur la dynamique arithmétique des automorphismes de l’espace affine