Analytic theory of curvettes and dicriticals
- 82 Downloads
The geometric theory of pencils in a germ of a smooth complex surface has been studied by several authors and their properties strongly lie on the structure of the dicritical components of their resolution, whereas the strict transform of the general element of the pencil by the resolution can be seen as a union of the so-called curvettes. This theory can be interpreted as the study of ideals (with two generators) in the ring of complex convergent power series in two variables, which is a local regular ring of dimension 2. In this work, we study properties of curvettes and dicriticals in an arbitrary local regular ring of dimension 2 (without restrictions on its characteristic or the one of its residue field). All the results are stated in purely algebraic terms though the ideas come from geometry.
KeywordsAnalytic Contact number Curvette Dicritical divisor
Mathematics Subject Classification (2000)Primary 14A05
Second author is partially supported by MTM2010-21740-C02-02.
- 2.Abhyankar, S.: Ramification theoretic methods in algebraic geometry. Annals of Mathematics Studies, no. 43. Princeton University Press, Princeton, NJ (1959)Google Scholar
- 3.Abhyankar, S.: Resolution of singularities of embedded algebraic surfaces. Pure and Applied Mathematics, Vol. 24, Academic Press, New York. Second Enlarged Edition of 1998 Published by Springer-Verlag (1966)Google Scholar
- 5.Abhyankar, S.: Desingularization of plane curves. In: Singularities, Part 1 (Arcata, Calif., 1981), Proc. Sympos. Pure Math., vol. 40, pp. 1–45. Amer. Math. Soc., Providence, RI (1983)Google Scholar
- 8.Abhyankar, S.: Lectures on algebra. Vol. I. World Scientific Publishing Co., Pte. Ltd., Hackensack, NJ (2006)Google Scholar
- 10.Abhyankar, S.: Inversion and invariance of characteristic terms: Part I. In: The legacy of Alladi Ramakrishnan in the mathematical sciences, pp. 93–168. Springer, New York (2010).Google Scholar
- 25.Nagata, M.: Local rings. Interscience Tracts in Pure and Applied Mathematics, No. 13. Interscience Publishers a division of John Wiley & Sons, New York-London (1962)Google Scholar
- 26.Zariski, O., Samuel, P.: Commutative algebra. Vol. 1. Springer-Verlag, New York (1975). With the cooperation of I. S. Cohen, Corrected reprinting of the 1958 edition, Graduate Texts in Mathematics, No. 28Google Scholar
- 27.Zariski, O., Samuel, P.: Commutative algebra. Vol. II. Springer, New York (1975). (Reprint of the 1960 edition, Graduate Texts in Mathematics, Vol. 29)Google Scholar