Quadratic transforms inside their generic incarnations
Author:
Shreeram S. Abhyankar
Journal:
Proc. Amer. Math. Soc. 140 (2012), 41114126
MSC (2010):
Primary 14A05
Published electronically:
April 12, 2012
MathSciNet review:
2957201
Fulltext 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.
 [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,
SpringerVerlag, 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), 10.1090/S0002993998041835
 [Ab8]
S.
S. Abhyankar, Lectures on algebra. Vol. I, World Scientific
Publishing Co. Pte. Ltd., Hackensack, NJ, 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), 10.1007/9781441962638_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), 10.1007/s132260100017x
 [Ab11]
Shreeram
S. Abhyankar, Pillars and towers of quadratic
transformations, Proc. Amer. Math. Soc.
139 (2011), no. 9,
3067–3082. MR 2811263
(2012f:14003), 10.1090/S000299392011107317
 [Ab12]
Shreeram
S. Abhyankar, More about dicriticals, Proc. Amer. Math. Soc. 139 (2011), no. 9, 3083–3097. MR 2811264
(2012d:13047), 10.1090/S000299392011107329
 [Nag]
Masayoshi
Nagata, Local rings, Interscience Tracts in Pure and Applied
Mathematics, No. 13, Interscience Publishers a division of John Wiley &
Sons New YorkLondon, 1962. MR 0155856
(27 #5790)
 [Ab1]
 S. S. Abhyankar, On the valuations centered in a local domain, American Journal of Mathematics, vol. 78 (1956), pages 321348. 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 SpringerVerlag. MR 1617523 (99c:14021)
 [Ab4]
 S. S. Abhyankar, Historical ramblings in algebraic geometry and related algebra, American Mathematical Monthly, vol. 83 (1976), pages 409448. 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 145. 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 15831596. 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 93168, 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 7797. MR 2650101 (2011d:14107)
 [Ab11]
 S. S. Abhyankar, Pillars and towers of quadratic transformations, Proceedings of the American Mathematical Society, vol. 139 (2011), pages 30673082. MR 2811263
 [Ab12]
 S. S. Abhyankar, More about dicriticals, Proceedings of the American Mathematical Society, vol. 139 (2011), pages 30833097. 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:
http://dx.doi.org/10.1090/S000299392012115837
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.
