Low pole order frames on vertical jets of the universal hypersurface

Merker, Joël

Low pole order frames on vertical jets of the universal hypersurface
[Repères méromorphes sur les jets de l’hypersurface universelle]

Merker, Joël

Annales de l'Institut Fourier, Tome 59 (2009), p. 1077-1104 / Harvested from Numdam

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Résumé
Abstract

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

𝒳 \subset ℙ^{n + 1} \times ℙ^{\frac{(n + 1 + d)!}{(n + 1)! d!} - 1}

qui paramétrise toutes les hypersurfaces projectives $X \subset ℙ^{n + 1} (ℂ)$ de degré $d$ . Siu a annoncé en 2004 que, pour $k = n$ , il existe deux constantes $c_{n} \geq 1$ et $c_{n}^{'} \geq 1$ telles que le fibré tangent tensorisé

T_{J_{vert}^{n} (𝒳)} \otimes 𝒪_{ℙ^{n + 1}} (c_{n}) \otimes 𝒪_{ℙ^{\frac{(n + 1 + d)!}{((n + 1)! d!)} - 1}} (c_{n}^{'})

est engendré par ses sections globales. Nous établissons cette propriété hors d’un certain ensemble algébrique exceptionnel $Σ \subset J_{vert}^{n} (𝒳)$ défini par l’annulation de certains wronskiens, avec l’ordre de pôles effectif $c_{n} = \frac{1}{2} (n^{2} + 5 n)$ , retrouvant ainsi $c_{2} = 7$ (Paŭn), $c_{3} = 12$ (Rousseau), et avec $c_{n}^{'} = 1$ .

De plus, quitte à augmenter $c_{n}$ jusqu’à $c_{n} = n^{2} + 2 n$ , la même propriété d’engendrement est satisfaite hors du plus petit sous-ensemble $\tilde{Σ} \subset Σ \subset J_{vert}^{n} (𝒳)$ 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 $\tilde{Σ}$ ) des courbes holomorphes entières $ℂ \to X$ en découleront prochainement.

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

𝒳 \subset ℙ^{n + 1} \times ℙ^{\frac{(n + 1 + d)!}{((n + 1)! d!)} - 1}

parametrizing all projective hypersurfaces $X \subset ℙ^{n + 1} (ℂ)$ of degree $d$ . In 2004, for $k = n$ , Siu announced that there exist two constants $c_{n} \geq 1$ and $c_{n}^{'} \geq 1$ such that the twisted tangent bundle

T_{J_{vert}^{n} (𝒳)} \otimes 𝒪_{ℙ^{n + 1}} (c_{n}) \otimes 𝒪_{ℙ^{\frac{(n + 1 + d)!}{((n + 1)! d!)} - 1}} (c_{n}^{'})

is generated at every point by its global sections. In the present article, we establish this property outside a certain exceptional algebraic subset $Σ \subset J_{vert}^{n} (𝒳)$ defined by the vanishing of certain Wronskians, with the effective pole order $c_{n} = \frac{1}{2} (n^{2} + 5 n)$ , 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 $c_{n} = n^{2} + 2 n$ , the same generation property holds outside the smaller set $\tilde{Σ} \subset Σ \subset J_{vert}^{n} (𝒳)$ which is defined by the vanishing of all first order jets. Applications to weak (with $Σ$ ) and to strong (with $\tilde{Σ}$ ) algebraic degeneracy of entire holomorphic curves $ℂ \to X$ are upcoming.

Publié le : 2009-01-01

MR 2543663 | Zbl 1172.32005

DOI : https://doi.org/10.5802/aif.2458
Classification: 32Q45, 14N05, 14J70
Mots 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

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     author = {Merker, Jo\"el},
     title = {Low pole order frames on vertical jets of the universal hypersurface},
     journal = {Annales de l'Institut Fourier},
     volume = {59},
     year = {2009},
     pages = {1077-1104},
     doi = {10.5802/aif.2458},
     zbl = {1172.32005},
     mrnumber = {2543663},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 59 (2009) pp. 1077-1104. doi : 10.5802/aif.2458. http://gdmltest.u-ga.fr/item/AIF_2009__59_3_1077_0/

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