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Exotic embeddings of open subsets of affine space

Author: Zbigniew Jelonek
Journal: Proc. Amer. Math. Soc. 140 (2012), 1561-1568
MSC (2010): Primary 14R10
Posted: September 12, 2011
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Abstract: We show that every Zariski open affine proper subset $X\subset\mathbb{A}^n$ has infinitely many non-equivalent embeddings into $\mathbb{A}^{n+1}$ . Moreover, we give some examples of non-equivalent embeddings of such in higher codimension.

References

Bibliography

1. Shreeram Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956), 321–348. MR 0082477 (18,556b)
2. Shreeram S. Abhyankar and Tzuong Tsieng Moh, Embeddings of the line in the plane, J. Reine Angew. Math. 276 (1975), 148–166. MR 0379502 (52 #407)
3. Harm Derksen, Frank Kutzschebauch, and Jörg Winkelmann, Subvarieties of 𝐶ⁿ with non-extendable automorphisms, J. Reine Angew. Math. 508 (1999), 213–235. MR 1676877 (2000c:32072), http://dx.doi.org/10.1515/crll.1999.028
4. Zbigniew Jelonek, The extension of regular and rational embeddings, Math. Ann. 277 (1987), no. 1, 113–120. MR 884649 (88e:14016), http://dx.doi.org/10.1007/BF01457281
5. Zbigniew Jelonek, Identity sets for polynomial automorphisms, J. Pure Appl. Algebra 76 (1991), no. 3, 333–337. MR 1147306 (93c:14018), http://dx.doi.org/10.1016/0022-4049(91)90141-N
6. Zbigniew Jelonek, Irreducible identity sets for polynomial automorphisms, Math. Z. 212 (1993), no. 4, 601–617. MR 1214049 (94c:14014), http://dx.doi.org/10.1007/BF02571679
7. Zbigniew Jelonek, A note about the extension of polynomial embeddings, Bull. Polish Acad. Sci. Math. 43 (1995), no. 3, 239–244. MR 1415002 (97i:14011)
8. Zbigniew Jelonek, A hypersurface which has the Abhyankar-Moh property, Math. Ann. 308 (1997), no. 1, 73–84. MR 1446200 (98d:14019), http://dx.doi.org/10.1007/s002080050065
9. Zbigniew Jelonek, Exotic embeddings of smooth affine varieties, J. Algebra 297 (2006), no. 2, 530–541. MR 2209273 (2006m:14084), http://dx.doi.org/10.1016/j.jalgebra.2005.11.011
10. Zbigniew Jelonek, Manifolds with a unique embedding, Colloq. Math. 117 (2009), no. 2, 299–317. MR 2550135 (2010m:58003), http://dx.doi.org/10.4064/cm117-2-13
11. Shulim Kaliman, Extensions of isomorphisms between affine algebraic subvarieties of 𝑘ⁿ to automorphisms of 𝑘ⁿ, Proc. Amer. Math. Soc. 113 (1991), no. 2, 325–334. MR 1076575 (91m:14019), http://dx.doi.org/10.1090/S0002-9939-1991-1076575-3
12. 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 (2000f:14099), http://dx.doi.org/10.1007/BF01236662
13. T. Y. Lam, Serre’s conjecture, Lecture Notes in Mathematics, Vol. 635, Springer-Verlag, Berlin, 1978. MR 0485842 (58 #5644)
14. Michèle Raynaud, Modules projectifs universels, Invent. Math. 6 (1968), 1–26 (French). MR 0236164 (38 #4462)
15. Vladimir Shpilrain and Jie-Tai Yu, Embeddings of hypersurfaces in affine spaces, J. Algebra 239 (2001), no. 1, 161–173. MR 1827879 (2002m:14051), http://dx.doi.org/10.1006/jabr.2000.8677
16. V. Srinivas, On the embedding dimension of an affine variety, Math. Ann. 289 (1991), no. 1, 125–132. MR 1087241 (92a:14013), http://dx.doi.org/10.1007/BF01446563
17. Masakazu Suzuki, Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l’espace 𝐶², J. Math. Soc. Japan 26 (1974), 241–257 (French). MR 0338423 (49 #3188)
18. Richard G. Swan, Vector bundles, projective modules and the 𝐾-theory of spheres, Algebraic topology and algebraic 𝐾-theory (Princeton, N.J., 1983), Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 432–522. MR 921488 (89f:18009)
19. Burt Totaro, The automorphism group of an affine quadric, Math. Proc. Cambridge Philos. Soc. 143 (2007), no. 1, 1–8. MR 2340971 (2008j:11050), http://dx.doi.org/10.1017/S0305004107000357

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