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 which take into account the inflection points of the fibres of . 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.
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 un germe de fonction holomorphe avec un point critique à l’origine et son hessien. On étudie l’application en oubliant les relations différentielles entre et . Ceci permet de définir une notion d’équivalence par rapport aux inflexions appelée -équivalence ainsi qu’une notion de déformation verselle par rapport aux inflexions. On montre qu’il existe un seul germe -stable puis on donne la liste des fonctions -simples. À l’aide des techniques introduites, on détermine la stratification par rapport aux inflexions de l’espace des déformations d’un germe -simple.
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
Garay, Mauricio 1
@article{AIF_2002__52_3_849_0, author = {Garay, Mauricio}, title = {On vanishing inflection points of plane curves}, journal = {Annales de l'Institut Fourier}, pages = {849--880}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {52}, number = {3}, year = {2002}, doi = {10.5802/aif.1904}, zbl = {1116.14301}, mrnumber = {1907390}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1904/} }
TY - JOUR AU - Garay, Mauricio TI - On vanishing inflection points of plane curves JO - Annales de l'Institut Fourier PY - 2002 SP - 849 EP - 880 VL - 52 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1904/ DO - 10.5802/aif.1904 LA - en ID - AIF_2002__52_3_849_0 ER -
%0 Journal Article %A Garay, Mauricio %T On vanishing inflection points of plane curves %J Annales de l'Institut Fourier %D 2002 %P 849-880 %V 52 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1904/ %R 10.5802/aif.1904 %G en %F AIF_2002__52_3_849_0
Garay, Mauricio. On vanishing inflection points of plane curves. Annales de l'Institut Fourier, Volume 52 (2002) no. 3, pp. 849-880. doi : 10.5802/aif.1904. https://aif.centre-mersenne.org/articles/10.5802/aif.1904/
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