Journal of Singularities
volume 9 (2014), 82-100
Received 3 April 2012. Received in revised form 16 December 2013.
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).
holomorphic foliation, Morse center
Mathematical Subject Classification:
A. Lins Neto
Instituto de Matemática Pura e Aplicada
Estrada Dona Castorina, 110
Horto, Rio de Janeiro, Brasil