Infinitely near imposed singularities and singularities of polar curves. (English) Zbl 0675.14009
Let \(\xi\) be a complex plane algebroid curve, defined by the equation \(f=0\). This paper is mainly devoted to the study of the singularities of the polar curves of \(\xi\) (defined by \(a\partial f/\partial x+b\partial f/\partial y=0\), a and b being complex numbers). The main results are theorem 8.1 and 11.3.
Theorem 8.1 gives the virtual behaviour of the polar curves and is a precise version of the old claim (Noether) saying that the polar curves have a (n-1)-fold point at each ordinary or infinitely near n-fold point of \(\xi\).
Theorem 11.3 describes the singularities, and in particular the topological types, of general and special polar curves of \(\xi\), assuming \(\xi\) to be unibranched with given characteristic exponents and generic Puiseux coefficients.
The §§ 2 to 6 are devoted to developing a theory of infinitely near imposed singularities which is needed for polar curves. § 12 gives an account of classical work on the subject.
Theorem 8.1 gives the virtual behaviour of the polar curves and is a precise version of the old claim (Noether) saying that the polar curves have a (n-1)-fold point at each ordinary or infinitely near n-fold point of \(\xi\).
Theorem 11.3 describes the singularities, and in particular the topological types, of general and special polar curves of \(\xi\), assuming \(\xi\) to be unibranched with given characteristic exponents and generic Puiseux coefficients.
The §§ 2 to 6 are devoted to developing a theory of infinitely near imposed singularities which is needed for polar curves. § 12 gives an account of classical work on the subject.
Reviewer: E.Casas-Alvero
MSC:
| 14H20 | Singularities of curves, local rings |
| 14H45 | Special algebraic curves and curves of low genus |
| 14B10 | Infinitesimal methods in algebraic geometry |
Keywords:
complex plane algebroid curve; singularities of polar curves; topological type; infinitely near imposed singularitiesReferences:
| [1] | Bertini, E.: Geometria proiettiva degli ipespazi. Messina: Principato, 1923 · JFM 49.0484.08 |
| [2] | Casas-Alvero, E.: Moduli of algebroid plane curves, in: Algebraic geometry, La R?bida 1981 (Lect. Notes Math., vol. 961, pp. 32-83.) Berlin Heidelberg New York: Springer 1983 |
| [3] | Casas-Alvero, E.: On the singularities of polar curves. Manuscr. Math.43, 167-190 (1983) · Zbl 0526.14015 · doi:10.1007/BF01165829 |
| [4] | Coolidge, J.L.: A treatise on algebraic plane curves. Oxford: Oxford University Press 1931 · JFM 57.0820.06 |
| [5] | Enriques, F.: Teoria geometrica delle equazioni e delle funzioni algebriche. Bologna: Zanichelli 1915 · JFM 45.1356.02 |
| [6] | Ephraim, R.: Special polars and curves with one place at infinity. Proc. Symp. Pure Math.40.1, 353-361 (1983) · Zbl 0537.14020 |
| [7] | Hardy, G.H., Wright, E. M.: An introduction to the theory of numbers. Oxford: Clarendon Press 1968 · Zbl 0020.29201 |
| [8] | Kuo, T.C., Lu, Y.C.: On analytic function germs of two variables. Topology16, 299-310 (1977) · Zbl 0378.32001 · doi:10.1016/0040-9383(77)90037-4 |
| [9] | Merle, M.: Invariants polaires des courbes planes. Invent. Math.41, 103-111 (1977) · Zbl 0371.14003 · doi:10.1007/BF01418370 |
| [10] | Noether, M.: Rationale Ausfuhrung der Operationen... Math. Ann.23, 311-358 (1884) · JFM 16.0349.01 · doi:10.1007/BF01446394 |
| [11] | Northcott, D.G.: On the notion of a first neighbourhood ring... Proc. Cambridge Phil. Soc.53, 43-56 (1957) · Zbl 0082.03304 · doi:10.1017/S0305004100031972 |
| [12] | Pham, F., Deformations equisinguli?res des id?aux jacobiens des courbes planes, in: Proc of Liverpool Singularities Symposium II (Lect. Notes Math., vol. 209, pp. 218-233) Berlin Heidelberg New York: Springer 1971 |
| [13] | Segre, B., Sullo scioglimento delle singolarit? delle variet? algebriche. Ann. Mat. Pura Appl. IV.33, 5-48 (1952) · Zbl 0046.38901 |
| [14] | Semple, J.G., Kneebone, G.T.: Algebraic curves. London: Oxford University Press 1959 · Zbl 0105.34301 |
| [15] | Severi, F.: Tratatto di geometria algebrica. Bologna: Zanichelli 1926 · JFM 52.0650.01 |
| [16] | Teissier, B.: Vari?t?s polaires I. Invariants polaires des singularit?s d’hypersurfaces. Invent. Math.40, 267-292 (1977) · Zbl 0446.32002 · doi:10.1007/BF01425742 |
| [17] | Teissier, B.: The hunting of invariants in the geometry of discriminants, in: Real and complex singularities?Oslo 1976, pp. 565-677. Sijthoff-Noordhoff 1977 |
| [18] | Waerden, B. L. Van der: Infinitely near points. Ind. Math.12, 401-410 (1950) · Zbl 0038.32101 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
