Low pole order frames on vertical jets of the universal hypersurface
Annales de l'Institut Fourier, Volume 59 (2009) no. 3, p. 1077-1104

For low order jets, it is known how to construct meromorphic frames on the space of the so-called vertical k-jets J vert k (𝒳) of the universal hypersurface

𝒳n+1×(n+1+d)!((n+1)!d!)-1

parametrizing all projective hypersurfaces X n+1 () of degree d. In 2004, for k=n, Siu announced that there exist two constants c n 1 and c n 1 such that the twisted tangent bundle

TJvertn(𝒳)𝒪n+1(cn)𝒪(n+1+d)!((n+1)!d!)-1(cn)

is generated at every point by its global sections. In the present article, we establish this property outside a certain exceptional algebraic subset ΣJ vert n (𝒳) defined by the vanishing of certain Wronskians, with the effective pole order c n =1 2(n 2 +5n), thus recovering c 2 =7 (Paŭn), c 3 =12 (Rousseau), and with c n =1.

Moreover, at the cost of raising c n up to c n =n 2 +2n, the same generation property holds outside the smaller set Σ ˜ΣJ vert n (𝒳) which is defined by the vanishing of all first order jets. Applications to weak (with Σ) and to strong (with Σ ˜) algebraic degeneracy of entire holomorphic curves X are upcoming.

Pour des ordres de jets petits, on sait construire des repères méromorphes sur l’espace des jets verticaux J vert k (𝒳) de l’hypersurface universelle

𝒳n+1×(n+1+d)!(n+1)!d!-1

qui paramétrise toutes les hypersurfaces projectives X n+1 () de degré d. Siu a annoncé en 2004 que, pour k=n, il existe deux constantes c n 1 et c n 1 telles que le fibré tangent tensorisé

TJvertn(𝒳)𝒪n+1(cn)𝒪(n+1+d)!((n+1)!d!)-1(cn)

est engendré par ses sections globales. Nous établissons cette propriété hors d’un certain ensemble algébrique exceptionnel ΣJ vert n (𝒳) défini par l’annulation de certains wronskiens, avec l’ordre de pôles effectif c n =1 2(n 2 +5n), retrouvant ainsi c 2 =7 (Paŭn), c 3 =12 (Rousseau), et avec c n =1.

De plus, quitte à augmenter c n jusqu’à c n =n 2 +2n, la même propriété d’engendrement est satisfaite hors du plus petit sous-ensemble Σ ˜ΣJ vert n (𝒳) qui est défini par l’annulation de tous les jets d’ordre 1. Des applications à la dégénérescence algébrique faible (avec Σ) et forte (avec Σ ˜) des courbes holomorphes entières X en découleront prochainement.

DOI : https://doi.org/10.5802/aif.2458
Classification:  32Q45,  14N05,  14J70
Keywords: Multivariate Faà di Bruno formula, projective algebraic hypersurfaces, jets of holomorphic curves, weak and strong Green-Griffiths algebraic degeneracy
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {3},
     year = {2009},
     pages = {1077-1104},
     doi = {10.5802/aif.2458},
     mrnumber = {2543663},
     zbl = {1172.32005},
     language = {en},
     url = {http://www.numdam.org/item/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/item/AIF_2009__59_3_1077_0/

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