Low pole order frames on vertical jets of the universal hypersurface

Merker, Joël

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

Pour des ordres de jets petits, on sait construire des repères méromorphes sur l’espace des jets verticaux $J_{\mathsf{\text{vert}}}^k\left(𝒳\right)$ de l’hypersurface universelle

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

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

$$T_{J_{\mathsf{\text{vert}}}^n\left(𝒳\right)}\otimes {𝒪}_{{\mathbb{P}}^{n+1}}\left(c_n\right)\otimes {𝒪}_{{\mathbb{P}}^{\frac{(n+1+d)!}{\left(\right(n+1)!d!)}-1}}\left(c_n^{\prime }\right)$$

est engendré par ses sections globales. Nous établissons cette propriété hors d’un certain ensemble algébrique exceptionnel $\Sigma \subset J_{\mathsf{\text{vert}}}^n\left(𝒳\right)$ défini par l’annulation de certains wronskiens, avec l’ordre de pôles effectif $c_n=\frac{1}{2}\left(n^2+5n\right)$, retrouvant ainsi $c_2=7$ (Paŭn), $c_3=12$ (Rousseau), et avec $c_n^{\prime }=1$.

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 $\tilde{\Sigma }\subset \Sigma \subset J_{\mathsf{\text{vert}}}^n\left(𝒳\right)$ qui est défini par l’annulation de tous les jets d’ordre $1$. Des applications à la dégénérescence algébrique faible (avec $\Sigma$) et forte (avec $\tilde{\Sigma }$) 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 $J_{\mathsf{\text{vert}}}^k\left(𝒳\right)$ of the universal hypersurface

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

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

$$T_{J_{\mathsf{\text{vert}}}^n\left(𝒳\right)}\otimes {𝒪}_{{\mathbb{P}}^{n+1}}\left(c_n\right)\otimes {𝒪}_{{\mathbb{P}}^{\frac{(n+1+d)!}{\left(\right(n+1)!d!)}-1}}\left(c_n^{\prime }\right)$$

is generated at every point by its global sections. In the present article, we establish this property outside a certain exceptional algebraic subset $\Sigma \subset J_{\mathsf{\text{vert}}}^n\left(𝒳\right)$ defined by the vanishing of certain Wronskians, with the effective pole order $c_n=\frac{1}{2}\left(n^2+5n\right)$, thus recovering $c_2=7$ (Paŭn), $c_3=12$ (Rousseau), and with $c_n^{\prime }=1$.

Moreover, at the cost of raising $c_n$ up to $c_n=n^2+2n$, the same generation property holds outside the smaller set $\tilde{\Sigma }\subset \Sigma \subset J_{\mathsf{\text{vert}}}^n\left(𝒳\right)$ which is defined by the vanishing of all first order jets. Applications to weak (with $\Sigma$) and to strong (with $\tilde{\Sigma }$) algebraic degeneracy of entire holomorphic curves $ℂ\to X$ are upcoming.

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

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