Foliations with a Morse center
Journal of Singularities, Tome 9 (2014), pp. 82-100
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We say that a holomorphic foliation F on a complex surface M has a Morse center at p in M if F has a local first integral with a Morse singularity at p. Given a line bundle L on M, let Fol(M,L) be the set of foliations F on M such that T^*(F)=L, and let Fol_C(M,L) be the closure of the set of F in Fol(M,L) such that F has a Morse center. In the first result of this paper we prove that Fol_C(M,L) is an algebraic subset of Fol(M,L). We apply this result to prove the persistence of more than one Morse center for some known examples, as for instance the logarithmic and pull-back foliations. As an application, we give a simple proof that R(1,d+1) is an irreducible component of the space of foliations of degree d with a Morse center on P^2, where R(m,n) denotes the space of foliations with a rational first integral of the form f^m/g^n with m dg(f)=n dg(g).
@article{10_5427_jsing_2014_9h,
author = {A. Lins Neto},
title = {Foliations with a {Morse} center},
journal = {Journal of Singularities},
pages = {82--100},
publisher = {mathdoc},
volume = {9},
year = {2014},
doi = {10.5427/jsing.2014.9h},
url = {http://geodesic.mathdoc.fr/articles/10.5427/jsing.2014.9h/}
}
A. Lins Neto. Foliations with a Morse center. Journal of Singularities, Tome 9 (2014), pp. 82-100. doi: 10.5427/jsing.2014.9h
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