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)

 
 

 

Quadratic transforms inside their generic incarnations


Author: Shreeram S. Abhyankar
Journal: Proc. Amer. Math. Soc. 140 (2012), 4111-4126
MSC (2010): Primary 14A05
DOI: https://doi.org/10.1090/S0002-9939-2012-11583-7
Published electronically: April 12, 2012
MathSciNet review: 2957201
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The generic incarnation of a quadratic transform is obtained by adjoining an indeterminate. This leads to the singularity tree of a generic member of a pencil of curves.


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

  • [Ab1] S. S. Abhyankar, On the valuations centered in a local domain, American Journal of Mathematics, vol. 78 (1956), pages 321-348. MR 0082477 (18:556b)
  • [Ab2] S. S. Abhyankar, Ramification Theoretic Methods in Algebraic Geometry, Princeton University Press, 1959. MR 0105416 (21:4158)
  • [Ab3] S. S. Abhyankar, Resolution of Singularities of Embedded Algebraic Surfaces, First Edition of 1966, Published by Academic Press; Second Enlarged Edition of 1998, Published by Springer-Verlag. MR 1617523 (99c:14021)
  • [Ab4] S. S. Abhyankar, Historical ramblings in algebraic geometry and related algebra, American Mathematical Monthly, vol. 83 (1976), pages 409-448. MR 0401754 (53:5581)
  • [Ab5] S. S. Abhyankar, Desingularization of plane curves, Singularities, Proceedings of Symposia in Pure Mathematics, American Mathematical Society, vol. 40, Part 1 (1983), pages 1-45. MR 713043 (85d:14024)
  • [Ab6] S. S. Abhyankar, Algebraic Geometry for Scientists and Engineers, American Mathematical Society, 1990. MR 1075991 (92a:14001)
  • [Ab7] S. S. Abhyankar, Polynomial expansion, Proceedings of the American Mathematical Society, vol. 126 (1998), pages 1583-1596. MR 1443142 (98g:12003)
  • [Ab8] S. S. Abhyankar, Lectures on Algebra I, World Scientific, 2006. MR 2253320 (2007k:13001)
  • [Ab9] S. S. Abhyankar, Inversion and Invariance of Characteristic Terms Part I, The Legacy of Alladi Ramakrishnan in the Mathematical Sciences (2010), pages 93-168, Springer. MR 2744259 (2012b:14057)
  • [Ab10] S. S. Abhyankar, Dicritical divisors and Jacobian problem, Indian Journal of Pure and Applied Mathematics, vol. 41 (2010), pages 77-97. MR 2650101 (2011d:14107)
  • [Ab11] S. S. Abhyankar, Pillars and towers of quadratic transformations, Proceedings of the American Mathematical Society, vol. 139 (2011), pages 3067-3082. MR 2811263
  • [Ab12] S. S. Abhyankar, More about dicriticals, Proceedings of the American Mathematical Society, vol. 139 (2011), pages 3083-3097. MR 2811264
  • [Nag] M. Nagata, Local Rings, Wiley, 1962. MR 0155856 (27:5790)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 14A05

Retrieve articles in all journals with MSC (2010): 14A05


Additional Information

Shreeram S. Abhyankar
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: ram@cs.purdue.edu

DOI: https://doi.org/10.1090/S0002-9939-2012-11583-7
Keywords: Pencil, singularity tree, generic member
Received by editor(s): May 25, 2011
Published electronically: April 12, 2012
Communicated by: Lev Borisov
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society