Let be a codimension-one foliation on : for each point we define as the order of the first non-zero jet of a holomorphic 1-form defining at . The singular set of is . We prove (main Theorem 1.2) that a foliation satisfying for all has a non-constant rational first integral. Using this fact we are able to prove that any foliation of degree-three on , with , is either the pull-back of a foliation on , or has a transverse affine structure with poles. This extends previous results for foliations of degree at most two.
@article{ASNSP_2013_5_12_1_1_0, author = {Cerveau, Dominique and Lins Neto, Alcides}, title = {A structural theorem for codimension-one foliations on $\protect \mathbb{P}^n$, $n\ge 3$, with an application to degree-three foliations}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 12}, number = {1}, year = {2013}, pages = {1-41}, zbl = {1267.32030}, mrnumber = {3088436}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2013_5_12_1_1_0} }
Cerveau, Dominique; Lins Neto, Alcides. A structural theorem for codimension-one foliations on $\protect \mathbb{P}^n$, $n\ge 3$, with an application to degree-three foliations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 1, pp. 1-41. http://www.numdam.org/item/ASNSP_2013_5_12_1_1_0/
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