On vanishing inflection points of plane curves

Garay, Mauricio

Annales de l'Institut Fourier, Tome 52 (2002) no. 3, pp. 849-880.

Le but de cet article est d’introduire une théorie des formes normales et des déformations des courbes projectives planes qui tienne compte de leurs points d’inflexion. On procède de la façon suivante. Soit $f:({ℂ}^2,0)⟶(ℂ,0)$ un germe de fonction holomorphe avec un point critique à l’origine et ${\Delta }_f:({ℂ}^2,0)⟶(ℂ,0)$ son hessien. On étudie l’application $(f,{\Delta }_f)$ en oubliant les relations différentielles entre $f$ et ${\Delta }_f$. Ceci permet de définir une notion d’équivalence par rapport aux inflexions appelée $𝒫$-équivalence ainsi qu’une notion de déformation verselle par rapport aux inflexions. On montre qu’il existe un seul germe $𝒫$-stable puis on donne la liste des fonctions $𝒫$-simples. À l’aide des techniques introduites, on détermine la stratification par rapport aux inflexions de l’espace des déformations d’un germe $𝒫$-simple.

We study the local behaviour of inflection points of families of plane curves in the projective plane. We develop normal forms and versal deformation concepts for holomorphic function germs $f:({ℂ}^2,0)⟶(ℂ,0)$ which take into account the inflection points of the fibres of $f$. We give a classification of such function- germs which is a projective analog of Arnold’s A,D,E classification. We compute the versal deformation with respect to inflections of Morse function-germs.

[1] V.I. Arnold Normal forms for functions near degenerate critical points, the Weyl groups $A_k$, $D_k$, $E_k$ and Lagrangian singularities, Funct. Anal. Appl., Volume 6 (1972) no. 4, pp. 254-272 | DOI | MR | Zbl

[2] V.I. Arnold Mathematical methods of classical mechanics (Nauka) (1974 ; 1978), pp. 462 p | Zbl

[3] V.I. Arnold Wave front evolution and the equivariant Morse lemma, Comm. Pure Appl. Math., Volume 29 (1976) no. 6, pp. 557-582 | DOI | MR | Zbl

[4] V.I. Arnold Vanishing inflexions, Funct. Anal. Appl., Volume 18 (1984) no. 2, pp. 51-52 | MR | Zbl

[5] V.I. Arnold; V.I. Vassiliev; V.V. Goryunov; O.V. Lyashko Singularity theory I, dynamical systems VI, VINITI, Moscow (1993), pp. 245 p

[6] J. Damon The unfolding and determinacy theorems for subgroups of $𝒜$ and $𝒦$, Memoirs of the AMS, Volume vol. 50 (1984) | MR | Zbl

[7] M. Garay Théorie des points d'aplatissement évanescents des courbes planes et spatiales (2001) Thèse, Université de Paris 7, février (in English)

[8] M. Garay On simple families of functions (2002) (Preprint Mainz Universität)

[9] V.V. Goryunov Vector fields and functions on discriminants of complete intersections and bifurcation diagrams of projections, Funct. Anal. Appl., Volume 15 (1981), pp. 77-82 | Zbl

[10] P. Griffiths; J. Harris Principles of algebraic geometry, Wiley Interscience, 1978 | MR | Zbl

[11] M.E. Kazarian Singularities of the boundary of fundamental systems, flattening of projective curves, and Schubert cells, Itogi Nauli Tekh., Ser. Sovrem Probl. Math., Noviejshie dostizh., Volume 33 (1988), pp. 215-234 | Zbl

[11] M.E. Kazarian Singularities of the boundary of fundamental systems, flattening of projective curves, and Schubert cells, J. Soviet Math. (English transl.), Volume 52 (1990), pp. 3338-3349 | Zbl

[12] M.E. Kazarian Bifurcations of flattening and Schubert cells (Advances Soviet Math. I) (1990), pp. 145-156

[13] M.E. Kazarian Flattening of projective curves, singularities of Schubert stratifications of Grassmann and flag varieties, bifurcations of Weierstrass points of algebraic curves, Usp. Mat. Nauk, Volume 46 (1961) no. 5, pp. 79-119 | Zbl

[13] M.E. Kazarian Flattening of projective curves, singularities of Schubert stratifications of Grassmann and flag varieties, bifurcations of Weierstrass points of algebraic curves, Russ. Math. Surveys (English transl.), Volume 46 (1992) no. 5, pp. 91-136 | DOI | Zbl

[14] F. Klein Development of mathematics in the XIXth century, Lie groups history, frontiers and applications, Volume vol. 9 (1979), pp. 630 p. | Zbl

[15] J. Martinet Singularities of smooth functions and maps, London Math. Soc. Lecture Notes Series, vol. 58, Cambridge University Press, 1982 | MR | Zbl

[16] J. Mather Stability of $C^{\infty }$ mappings, I, Ann. of Math., Volume 87 (1968) | MR | Zbl

[16] J. Mather Stability of $C^{\infty }$ mappings, III, Pub. Sci. IHES, Volume 35 (1969) | Numdam | Zbl

[16] J. Mather Stability of $C^{\infty }$ mappings, II, Ann. of Math., Volume 89 (1969) | MR | Zbl

[16] J. Mather Stability of $C^{\infty }$ mappings, IV, Pub. Sci. IHES, Volume 37 (1970) | Numdam | Zbl

[16] J. Mather Stability of $C^{ä}$ mappings, V, Adv. in Math., Volume 4 (1970) | MR | Zbl

[16] J. Mather Stability of $C^{\infty }$ mappings, VI, Lecture Notes in Math., Volume 192 (1971) | DOI | MR | Zbl

[17] J. Plücker System der Analytischen Geometrie, Gessam. Wissen. Abhand., vol. Band 1 ; vol. 2, B.G. Teubner, 1834, 1898

[18] R. Piene Numerical characters of a curve in projective n-space, Real and complex singularities (1977), pp. 475-495 | Zbl

[19] G.N. Tyurina Locally semi-universal plane deformations of isolated singularities in complex space, Math. USSR Izv., Volume 32 (1968) no. 3, pp. 967-999 | Zbl

[20] R. Uribe Singularités symplectiques et de contact en géométrie différentielle des courbes et des surfaces (2001) (Thèse Université de Paris 7)

[21] V.M. Zakalyukin Lagrangian and Legendrian singularities, Funct. Anal. Appl., Volume 10 (1976) no. 1 | Zbl