Low pole order frames on vertical jets of the universal hypersurface

Merker, Joël

doi:10.5802/aif.2458

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

Merker, Joël ¹

¹ tabacckludge ’Ecole Normale Supérieure UMR 8553 du CNRS Département de Mathématiques et Applications 45 rue d’Ulm 75230 Paris Cedex 05 (France)

Annales de l'Institut Fourier, Tome 59 (2009) no. 3, pp. 1077-1104.

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.

MR Zbl

DOI : 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
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

Affiliations des auteurs :

Merker, Joël ¹

¹ tabacckludge ’Ecole Normale Supérieure UMR 8553 du CNRS Département de Mathématiques et Applications 45 rue d’Ulm 75230 Paris Cedex 05 (France)

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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},
     pages = {1077--1104},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {3},
     year = {2009},
     doi = {10.5802/aif.2458},
     mrnumber = {2543663},
     zbl = {1172.32005},
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Merker, Joël. Low pole order frames on vertical jets of the universal hypersurface. Annales de l'Institut Fourier, Tome 59 (2009) no. 3, pp. 1077-1104. doi : 10.5802/aif.2458. https://aif.centre-mersenne.org/articles/10.5802/aif.2458/

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