Quadratic transforms inside their generic incarnations

Author:
Shreeram S. Abhyankar

Journal:
Proc. Amer. Math. Soc. **140** (2012), 4111-4126

MSC (2010):
Primary 14A05

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.

**[Ab1]**Shreeram Abhyankar,*On the valuations centered in a local domain*, Amer. J. Math.**78**(1956), 321–348. MR**0082477****[Ab2]**Shreeram Abhyankar,*Ramification theoretic methods in algebraic geometry*, Annals of Mathematics Studies, no. 43, Princeton University Press, Princeton, N.J., 1959. MR**0105416****[Ab3]**S. S. Abhyankar,*Resolution of singularities of embedded algebraic surfaces*, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR**1617523****[Ab4]**Shreeram S. Abhyankar,*Historical ramblings in algebraic geometry and related algebra*, Amer. Math. Monthly**83**(1976), no. 6, 409–448. MR**0401754****[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****[Ab6]**Shreeram S. Abhyankar,*Algebraic geometry for scientists and engineers*, Mathematical Surveys and Monographs, vol. 35, American Mathematical Society, Providence, RI, 1990. MR**1075991****[Ab7]**Shreeram S. Abhyankar,*Polynomial expansion*, Proc. Amer. Math. Soc.**126**(1998), no. 6, 1583–1596. MR**1443142**, 10.1090/S0002-9939-98-04183-5**[Ab8]**S. S. Abhyankar,*Lectures on algebra. Vol. I*, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. MR**2253320****[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**, 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**, 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**, 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**, 10.1090/S0002-9939-2011-10732-9**[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**

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/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.