Quadratic transforms inside their generic incarnations

Abhyankar, Shreeram

doi:10.1090/S0002-9939-2012-11583-7

Quadratic transforms inside their generic incarnations
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Proc. Amer. Math. Soc. 140 (2012), 4111-4126 Request permission

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

Shreeram Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956), 321–348. MR 82477, DOI 10.2307/2372519
Shreeram Abhyankar, Ramification theoretic methods in algebraic geometry, Annals of Mathematics Studies, No. 43, Princeton University Press, Princeton, N.J., 1959. MR 0105416
S. S. Abhyankar, Resolution of singularities of embedded algebraic surfaces, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1617523, DOI 10.1007/978-3-662-03580-1
Shreeram S. Abhyankar, Historical ramblings in algebraic geometry and related algebra, Amer. Math. Monthly 83 (1976), no. 6, 409–448. MR 401754, DOI 10.2307/2318338
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
Shreeram S. Abhyankar, Algebraic geometry for scientists and engineers, Mathematical Surveys and Monographs, vol. 35, American Mathematical Society, Providence, RI, 1990. MR 1075991, DOI 10.1090/surv/035
Shreeram S. Abhyankar, Polynomial expansion, Proc. Amer. Math. Soc. 126 (1998), no. 6, 1583–1596. MR 1443142, DOI 10.1090/S0002-9939-98-04183-5
S. S. Abhyankar, Lectures on algebra. Vol. I, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. MR 2253320, DOI 10.1142/9789812773449
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, DOI 10.1007/978-1-4419-6263-8_{8}
Shreeram S. Abhyankar, Dicritical divisors and Jacobian problem, Indian J. Pure Appl. Math. 41 (2010), no. 1, 77–97. MR 2650101, DOI 10.1007/s13226-010-0017-x
Shreeram S. Abhyankar, Pillars and towers of quadratic transformations, Proc. Amer. Math. Soc. 139 (2011), no. 9, 3067–3082. MR 2811263, DOI 10.1090/S0002-9939-2011-10731-7
Shreeram S. Abhyankar, More about dicriticals, Proc. Amer. Math. Soc. 139 (2011), no. 9, 3083–3097. MR 2811264, DOI 10.1090/S0002-9939-2011-10732-9
Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856