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.

Classification : 37G25, 14N15
Keywords: Plücker formulas, normal forms, inflection points, bifurcation diagrams, projective geometry
Mot clés : formules de Plücker, formes normales, points d'inflexion, diagrammes de bifurcation, géométrie projective

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