Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Exotic embeddings of open subsets of affine space


Author: Zbigniew Jelonek
Journal: Proc. Amer. Math. Soc. 140 (2012), 1561-1568
MSC (2010): Primary 14R10
DOI: https://doi.org/10.1090/S0002-9939-2011-11037-2
Published electronically: September 12, 2011
MathSciNet review: 2869140
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $ X$ in higher codimension.


References [Enhancements On Off] (What's this?)

  • 1. Abhyankar, S., On the valuations centered in a local domain, Amer. J. Math. 78, 321-348 (1956). MR 0082477 (18:556b)
  • 2. Abhyankar, S., Moh, T., Embeddings of the line in the plane, J reine angew. Math. 276, 148-166 (1975). MR 0379502 (52:407)
  • 3. Derksen, H., Kutzsechebauch, F., Winkelmann, J., Subvarieties of $ \mathbb{C}^n$ with non-extendable automorphisms, J. Reine Angew. Math. 508, 213-235 (1999). MR 1676877 (2000c:32072)
  • 4. Jelonek, Z., The extension of regular and rational embeddings, Math. Ann. 277, 113-120 (1987). MR 884649 (88e:14016)
  • 5. Jelonek, Z., Identity sets for polynomial automorphisms, Journal of Pure & Applied Algebra 76, 333-337 (1991). MR 1147306 (93c:14018)
  • 6. Jelonek, Z., Irreducible identity sets for polynomial automorphisms, Math. Z. 212, 601-617 (1993). MR 1214049 (94c:14014)
  • 7. Jelonek, Z., A note about the extension of polynomial embeddings, Bull. Acad. Polon. Sci. Math. 43, 239-244 (1995). MR 1415002 (97i:14011)
  • 8. Jelonek, Z., A hypersurface which has the Abhyankar-Moh property, Math. Ann. 308, 73-84 (1997). MR 1446200 (98d:14019)
  • 9. Jelonek, Z., Exotic embeddings of smooth affine varieties, Journal of Algebra 297, 530-541 (2006). MR 2209273 (2006m:14084)
  • 10. Jelonek, Z., Manifolds with a unique embedding, Colloq. Math. 117, 299-317 (2009). MR 2550135 (2010m:58003)
  • 11. Kaliman, S., Extensions of isomorphisms between affine algebraic subvarieties of $ k^n$ to automorphisms of $ k^n$, Proc. Amer. Math. Soc. 113, 325-334 (1991). MR 1076575 (91m:14019)
  • 12. Kaliman, S., Zaidenberg, M., Affine modifications and affine hypersurfaces with a very transitive automorphism group, Transform. Groups 4, 53-95 (1999). MR 1669174 (2000f:14099)
  • 13. Lam, T., Serre's Conjecture, Lecture Notes in Math. 635, Springer-Verlag, 1978. MR 0485842 (58:5644)
  • 14. Raynaud, M., Modules projectifs universels, Invent. Math. 6, 1-26 (1968). MR 0236164 (38:4462)
  • 15. Shpilrain, V., Yu, J.T., Embeddings of hypersurfaces in affine spaces, J. of Alg. 239, 161-173 (2001). MR 1827879 (2002m:14051)
  • 16. Srinivas, V., On the embedding dimension of an affine variety, Math. Ann. 289, 125-132 (1991). MR 1087241 (92a:14013)
  • 17. Suzuki, M., Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l'espace $ \mathbb{C}^2,$ J. Math. Soc. Japan 26, 241-257 (1974). MR 0338423 (49:3188)
  • 18. Swan, R.G., Vector bundles, projective modules and the $ K$-theory of spheres, Algebraic topology and algebraic $ K$-theory (Princeton, N.J., 1983), 432-522, Ann. of Math. Studies 113, Princeton Univ. Press, 1987. MR 921488 (89f:18009)
  • 19. Totaro, B., The automorphism group of an affine quadric, Math. Proc. Camb. Phil. Soc. 143, 1-8 (2007). MR 2340971 (2008j:11050)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 14R10

Retrieve articles in all journals with MSC (2010): 14R10


Additional Information

Zbigniew Jelonek
Affiliation: Instytut Matematyczny, Polska Akademia Nauk, Śniadeckich, 00-956 Warszawa, Poland
Email: najelone@cyf-kr.edu.pl

DOI: https://doi.org/10.1090/S0002-9939-2011-11037-2
Keywords: Polynomial embedding, exotic embedding, Zariski open subset of affine space
Received by editor(s): January 16, 2011
Received by editor(s) in revised form: January 26, 2011
Published electronically: September 12, 2011
Additional Notes: The author was partially supported by the Polish Ministry of Science, grant 2010-2013
Communicated by: Lev Borisov
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society