We study codimension one foliations in projective space over by looking at its first order perturbations: unfoldings and deformations. We give special attention to foliations of rational and logarithmic type.
For a differential form defining a codimension one foliation, we present a graded module , related to the first order unfoldings of . If is a generic form of rational or logarithmic type, as a first application of the construction of , we classify the first order deformations that arise from first order unfoldings. Then, we count the number of isolated points in the singular set of , in terms of a Hilbert polynomial associated to .
We review the notion of regularity of in terms of a long complex of graded modules that we also introduce in this work. We use this complex to prove that, for generic rational and logarithmic foliations, is regular if and only if every unfolding is trivial up to isomorphism.
Nous étudions des feuilletages de codimension un dans l’espace projectif sur en regardant leurs perturbations du premier ordre : déploiements et déformations. Nous prêtons une attention particulière aux feuilletages rationnels et logarithmiques.
Pour une forme différentielle définissant un feuilletage de codimension un, nous présentons un module gradué , lié aux déploiements du premier ordre de . Si est une forme générique de type rationnel ou logarithmique, comme une première application de la construction de , nous classifions les déformations du premier ordre qui apparaissent à partir des déploiements du premier order. Ensuite, nous comptons le nombre de points isolés dans l’ensemble singulier de , en termes d’un polynôme de Hilbert associé à .
Nous revoyons la notion de régularité de en termes d’un complexe long de modules gradués que nous introduisons dans ce travail. Nous utilisons ce complexe pour prouver que, pour des feuilletages rationnels et logarithmiques génériques, est régulièr si et seulement si tout déploiement est trivial modulo isomorphisme.
Accepted:
Published online:
Keywords: foliations, codimension one, unfoldings, deformations
Mot clés : feuilletages, codimension un, déploiements, déformations
Molinuevo, Ariel 1
@article{AIF_2016__66_4_1583_0, author = {Molinuevo, Ariel}, title = {Unfoldings and deformations of rational and logarithmic foliations}, journal = {Annales de l'Institut Fourier}, pages = {1583--1613}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {66}, number = {4}, year = {2016}, doi = {10.5802/aif.3044}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3044/} }
TY - JOUR AU - Molinuevo, Ariel TI - Unfoldings and deformations of rational and logarithmic foliations JO - Annales de l'Institut Fourier PY - 2016 SP - 1583 EP - 1613 VL - 66 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3044/ DO - 10.5802/aif.3044 LA - en ID - AIF_2016__66_4_1583_0 ER -
%0 Journal Article %A Molinuevo, Ariel %T Unfoldings and deformations of rational and logarithmic foliations %J Annales de l'Institut Fourier %D 2016 %P 1583-1613 %V 66 %N 4 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3044/ %R 10.5802/aif.3044 %G en %F AIF_2016__66_4_1583_0
Molinuevo, Ariel. Unfoldings and deformations of rational and logarithmic foliations. Annales de l'Institut Fourier, Volume 66 (2016) no. 4, pp. 1583-1613. doi : 10.5802/aif.3044. https://aif.centre-mersenne.org/articles/10.5802/aif.3044/
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