For low order jets, it is known how to construct meromorphic frames on the space of the so-called vertical -jets of the universal hypersurface
parametrizing all projective hypersurfaces of degree . In 2004, for , Siu announced that there exist two constants and such that the twisted tangent bundle
is generated at every point by its global sections. In the present article, we establish this property outside a certain exceptional algebraic subset defined by the vanishing of certain Wronskians, with the effective pole order , thus recovering (Paŭn), (Rousseau), and with .
Moreover, at the cost of raising up to , the same generation property holds outside the smaller set which is defined by the vanishing of all first order jets. Applications to weak (with ) and to strong (with ) algebraic degeneracy of entire holomorphic curves are upcoming.
Pour des ordres de jets petits, on sait construire des repères méromorphes sur l’espace des jets verticaux de l’hypersurface universelle
qui paramétrise toutes les hypersurfaces projectives de degré . Siu a annoncé en 2004 que, pour , il existe deux constantes et telles que le fibré tangent tensorisé
est engendré par ses sections globales. Nous établissons cette propriété hors d’un certain ensemble algébrique exceptionnel défini par l’annulation de certains wronskiens, avec l’ordre de pôles effectif , retrouvant ainsi (Paŭn), (Rousseau), et avec .
De plus, quitte à augmenter jusqu’à , la même propriété d’engendrement est satisfaite hors du plus petit sous-ensemble qui est défini par l’annulation de tous les jets d’ordre . Des applications à la dégénérescence algébrique faible (avec ) et forte (avec ) des courbes holomorphes entières en découleront prochainement.
Keywords: Multivariate Faà di Bruno formula, projective algebraic hypersurfaces, jets of holomorphic curves, weak and strong Green-Griffiths algebraic degeneracy
Mot clés : formule de Faà di Bruno à plusieurs variables, hypersurfaces projectives algébriques, jets de courbes holomorphes, dégénérescence algébrique faible et forte au sens de Green-Griffiths
@article{AIF_2009__59_3_1077_0, author = {Merker, Jo\"el}, title = {Low pole order frames on vertical jets of the universal hypersurface}, journal = {Annales de l'Institut Fourier}, pages = {1077--1104}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {59}, number = {3}, year = {2009}, doi = {10.5802/aif.2458}, zbl = {1172.32005}, mrnumber = {2543663}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2458/} }
TY - JOUR AU - Merker, Joël TI - Low pole order frames on vertical jets of the universal hypersurface JO - Annales de l'Institut Fourier PY - 2009 SP - 1077 EP - 1104 VL - 59 IS - 3 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2458/ DO - 10.5802/aif.2458 LA - en ID - AIF_2009__59_3_1077_0 ER -
%0 Journal Article %A Merker, Joël %T Low pole order frames on vertical jets of the universal hypersurface %J Annales de l'Institut Fourier %D 2009 %P 1077-1104 %V 59 %N 3 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2458/ %R 10.5802/aif.2458 %G en %F AIF_2009__59_3_1077_0
Merker, Joël. Low pole order frames on vertical jets of the universal hypersurface. Annales de l'Institut Fourier, Volume 59 (2009) no. 3, pp. 1077-1104. doi : 10.5802/aif.2458. http://www.numdam.org/articles/10.5802/aif.2458/
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