ArticleOriginal scientific text
Title
Felix Klein's paper on real flexes vindicated
Authors 1
Affiliations
- Section de Mathématiques, Université de Genève, 2-4- rue du Lièvre, CH-1211 Genève 24, Switzerland
Abstract
In a paper written in 1876 [4], Felix Klein gave a formula relating the number of real flexes of a generic real plane projective curve to the number of real bitangents at non-real points and the degree, which shows in particular that the number of real flexes cannot exceed one third of the total number of flexes. We show that Klein's arguments can be made rigorous using a little of the theory of singularities of maps, justifying in particular his resort to explicit examples.
Bibliography
- S. Akbulut (ed.), Real Algebraic Geometry and Topology, Contemp. Math. 182, Amer. Math. Soc., Providence, 1996.
- R. Benedetti and R. Silhol, Spin and
- M. Golubitsky and V. Guillemin, Stable Mappings and their Singularities, Springer, New York-Heidelberg 1973.
- F. Klein, Eine neue Relation zwischen den Singularitäten einer algebraischen Curve, Math. Ann. 10 (1876), 199-209.
- I. R. Porteous, Simple Singularities of Maps, preprint, Columbia Univ., 1962, reprinted in: Proc. of the Liverpool Singularities Symp., Lecture Notes in Math. 192, Springer, Berlin, 1971, 217-236.
- F. Sottile, Enumerative Geometry for real Varieties, to appear.
- R. Thom, Stabilité structurelle et morphogénèse, W. A. Benjamin, Reading, 1972.
- O. Ya. Viro, Some integral calculus based on Euler characteristic, in: Topology and Geometry - Rohlin Seminar, Lecture Notes in Math. 1346, Springer, Berlin, 1988.
- C. T. C. Wall, Duality of real projective plane curves: Klein's equation, Topology 35 (1996), 355-362.
- H. G. Zeuthen, Sur les différentes formes des courbes planes du quatrième ordre, Math. Ann. 7 (1874), 410-432.
Pages:
195-210
Main language of publication
English
Published
1998
Exact and natural sciences