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)

 
 

 

Generic incarnations of quadratic transforms


Author: Shreeram S. Abhyankar
Journal: Proc. Amer. Math. Soc. 141 (2013), 4103-4117
MSC (2010): Primary 14A05
DOI: https://doi.org/10.1090/S0002-9939-2013-12027-7
Published electronically: August 7, 2013
MathSciNet review: 3105854
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The generic incarnation of a quadratic transform is obtained by adjoining an indeterminate. This is applied to proving some useful facts about intersection, such as the commutativity of contact numbers and the mixed characteristic generalization of Max Noether's Theorem on Refined Bezout.


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

  • [Ab1] Shreeram Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956), 321-348. MR 0082477 (18,556b)
  • [Ab2] Shreeram Abhyankar, Ramification theoretic methods in algebraic geometry, Annals of Mathematics Studies, no. 43, Princeton University Press, Princeton, N.J., 1959. MR 0105416 (21 #4158)
  • [Ab3] S. S. Abhyankar, Resolution of singularities of embedded algebraic surfaces, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1617523 (99c:14021)
  • [Ab4] Shreeram S. Abhyankar, Historical ramblings in algebraic geometry and related algebra, Amer. Math. Monthly 83 (1976), no. 6, 409-448. MR 0401754 (53 #5581)
  • [Ab5] Shreeram S. Abhyankar, Desingularization of plane curves, Singularities, Part 1 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 1-45. MR 713043 (85d:14024)
  • [Ab6] Shreeram S. Abhyankar, Algebraic geometry for scientists and engineers, Mathematical Surveys and Monographs, vol. 35, American Mathematical Society, Providence, RI, 1990. MR 1075991 (92a:14001)
  • [Ab7] Shreeram S. Abhyankar, Polynomial expansion, Proc. Amer. Math. Soc. 126 (1998), no. 6, 1583-1596. MR 1443142 (98g:12003), https://doi.org/10.1090/S0002-9939-98-04183-5
  • [Ab8] S. S. Abhyankar, Lectures on Algebra I, World Scientific, 2006. MR 2253320 (2007k:13001)
  • [Ab9] Shreeram S. Abhyankar, Inversion and invariance of characteristic terms: Part I, The legacy of Alladi Ramakrishnan in the mathematical sciences, Springer, New York, 2010, pp. 93-168. MR 2744259 (2012b:14057), https://doi.org/10.1007/978-1-4419-6263-8_8
  • [Ab10] Shreeram S. Abhyankar, Dicritical divisors and Jacobian problem, Indian J. Pure Appl. Math. 41 (2010), no. 1, 77-97. MR 2650101 (2011d:14107), https://doi.org/10.1007/s13226-010-0017-x
  • [Ab11] Shreeram S. Abhyankar, Pillars and towers of quadratic transformations, Proc. Amer. Math. Soc. 139 (2011), no. 9, 3067-3082. MR 2811263 (2012f:14003), https://doi.org/10.1090/S0002-9939-2011-10731-7
  • [Ab12] Shreeram S. Abhyankar, More about dicriticals, Proc. Amer. Math. Soc. 139 (2011), no. 9, 3083-3097. MR 2811264 (2012d:13047), https://doi.org/10.1090/S0002-9939-2011-10732-9
  • [Ab13] Shreeram S. Abhyankar, Quadratic transforms inside their generic incarnations, Proc. Amer. Math. Soc. 140 (2012), no. 12, 4111-4126. MR 2957201, https://doi.org/10.1090/S0002-9939-2012-11583-7
  • [Ab14] S. S. Abhyankar, Dicritical divisors and Dedekind's Gauss lemma, Proc. Amer. Math. Soc., to appear.
  • [AA1] S. S. Abhyankar and E. Artal Bartolo, Algebraic theory of curvettes and dicriticals, Proc. Amer. Math., to appear.
  • [AH1] Shreeram S. Abhyankar and William J. Heinzer, Existence of dicritical divisors, Amer. J. Math. 134 (2012), no. 1, 171-192. MR 2876143, https://doi.org/10.1353/ajm.2012.0002
  • [AH2] Shreeram S. Abhyankar and William J. Heinzer, Existence of dicritical divisors revisited, Proc. Indian Acad. Sci. Math. Sci. 121 (2011), no. 3, 267-290. MR 2867982 (2012h:14148), https://doi.org/10.1007/s12044-011-0035-6
  • [AH3] S. S. Abhyankar and W. J. Heinzer, Rees Valuations, Proc. Amer. Math. Soc., to appear.
  • [AL1] Shreeram S. Abhyankar and Ignacio Luengo, Algebraic theory of dicritical divisors, Amer. J. Math. 133 (2011), no. 6, 1713-1732. MR 2863374, https://doi.org/10.1353/ajm.2011.0045
  • [AL2] S. S. Abhyankar and I. Luengo, Spiders and dicriticals, Proc. Amer. Math. Soc., to appear.
  • [Nag] Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1962. MR 0155856 (27 #5790)
  • [Zar] O. Zariski and P. Samuel, Commutative Algebra, Springer-Verlag, 1975. MR 0384768 (52:5641)

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/S0002-9939-2013-12027-7
Keywords: Contact number, generic incarnation, mixed characteristic
Received by editor(s): February 1, 2012
Published electronically: August 7, 2013
Communicated by: Lev Borisov
Article copyright: © Copyright 2013 American Mathematical Society

American Mathematical Society