Low pole order frames on vertical jets of the universal hypersurface

Merker, Joël. "Low pole order frames on vertical jets of the universal hypersurface." Annales de l’institut Fourier 59.3 (2009): 1077-1104. <http://eudml.org/doc/10417>.

@article{Merker2009,
abstract = {For low order jets, it is known how to construct meromorphic frames on the space of the so-called vertical $k$-jets$J_\{\text\{\textsf \{vert\}\}\}^k (\mathcal\{X\})$ of the universal hypersurface\[ \mathcal\{X\} \subset \mathbb\{P\}^\{n+1\} \times \mathbb\{P\}^\{\frac\{(n+1+d)!\}\{((n+1)! d!)\}-1\} \]parametrizing all projective hypersurfaces $X \subset \mathbb\{P\}^\{n+1\} (\mathbb\{C\})$ 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\[ T\_\{J\_\{\text\{\textsf \{vert\}\}\}^n(\mathcal\{X\})\} \otimes \mathcal\{O\}\_\{\mathbb\{P\}^\{n+1\}\} (c\_n) \otimes \mathcal\{O\}\_\{\mathbb\{P\}^\{\frac\{(n+1+d)!\}\{((n+1)! d!)\}-1\}\} (c\_n^\{\prime\}) \]is generated at every point by its global sections. In the present article, we establish this property outside a certain exceptional algebraic subset $\Sigma \subset J_\{\text\{\textsf \{vert\}\}\}^n (\mathcal\{ X\})$ defined by the vanishing of certain Wronskians, with the effective pole order $c_n = \frac\{1\}\{2\}(\{ n^2 + 5n\})$, 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 $\widetilde\{\Sigma \} \subset \Sigma \subset J_\{\text\{\textsf \{vert\}\}\}^n (\mathcal\{ X\})$ which is defined by the vanishing of all first order jets. Applications to weak (with $\Sigma $) and to strong (with $\widetilde\{\Sigma \}$) algebraic degeneracy of entire holomorphic curves $\mathbb\{C\} \rightarrow X$ are upcoming.},
affiliation = {tabacckludge ’Ecole Normale Supérieure UMR 8553 du CNRS Département de Mathématiques et Applications 45 rue d’Ulm 75230 Paris Cedex 05 (France)},
author = {Merker, Joël},
journal = {Annales de l’institut Fourier},
keywords = {Multivariate Faà di Bruno formula; projective algebraic hypersurfaces; jets of holomorphic curves; weak and strong Green-Griffiths algebraic degeneracy; multivariate Faà di Bruno formula},
language = {eng},
number = {3},
pages = {1077-1104},
publisher = {Association des Annales de l’institut Fourier},
title = {Low pole order frames on vertical jets of the universal hypersurface},
url = {http://eudml.org/doc/10417},
volume = {59},
year = {2009},
}

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
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 3
SP - 1077
EP - 1104
AB - For low order jets, it is known how to construct meromorphic frames on the space of the so-called vertical $k$-jets$J_{\text{\textsf {vert}}}^k (\mathcal{X})$ of the universal hypersurface\[ \mathcal{X} \subset \mathbb{P}^{n+1} \times \mathbb{P}^{\frac{(n+1+d)!}{((n+1)! d!)}-1} \]parametrizing all projective hypersurfaces $X \subset \mathbb{P}^{n+1} (\mathbb{C})$ 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\[ T_{J_{\text{\textsf {vert}}}^n(\mathcal{X})} \otimes \mathcal{O}_{\mathbb{P}^{n+1}} (c_n) \otimes \mathcal{O}_{\mathbb{P}^{\frac{(n+1+d)!}{((n+1)! d!)}-1}} (c_n^{\prime}) \]is generated at every point by its global sections. In the present article, we establish this property outside a certain exceptional algebraic subset $\Sigma \subset J_{\text{\textsf {vert}}}^n (\mathcal{ X})$ defined by the vanishing of certain Wronskians, with the effective pole order $c_n = \frac{1}{2}({ n^2 + 5n})$, 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 $\widetilde{\Sigma } \subset \Sigma \subset J_{\text{\textsf {vert}}}^n (\mathcal{ X})$ which is defined by the vanishing of all first order jets. Applications to weak (with $\Sigma $) and to strong (with $\widetilde{\Sigma }$) algebraic degeneracy of entire holomorphic curves $\mathbb{C} \rightarrow X$ are upcoming.
LA - eng
KW - Multivariate Faà di Bruno formula; projective algebraic hypersurfaces; jets of holomorphic curves; weak and strong Green-Griffiths algebraic degeneracy; multivariate Faà di Bruno formula
UR - http://eudml.org/doc/10417
ER -