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Real logarithmic models for real analytic foliations in the plane Corral Pérez, Nuria Sanz Sánchez, Fernando Singular holomorphic foliation Logarithmic foliations Generalized curves Center-focus plane vector fields Let S be a germ of a holomorphic curve at (C2, 0) with finitely many branches S1, . . . , Sr and let I = (I1, . . . , Ir ) ∈ Cr . We show that there exists a nondicritical holomorphic foliation of logarithmic type at 0 ∈ C2 whose set of separatrices is S and having index Ii along Si in the sense of Lins Neto (Lecture Notes in Math. 1345, 192–232, 1988) if the following (necessary) condition holds: after a reduction of singularities π : M →(C2, 0) of S, the vector I gives rise, by the usual rules of transformation of indices by blowing-ups, to systems of indices along components of the total transform ¯S of S at points of the divisor E = π −1(0) satisfying: (a) at any singular point of ¯S the two indices along the branches of ¯S do not belong to Q≥0 and they are mutually inverse; (b) the sum of the indices along a component D of E for all points in D is equal to the self-intersection of D in M. This construction is used to show the existence of logarithmic models of real analytic foliations which are real generalized curves. Applications to real center-focus foliations are considered. Both authors were partially supported by the research project MTM2007-66262 (Ministerio de Ciencia e Innovación) and VA059A07 (Junta de Castilla y León). The first author was also partially supported by the research projects MTM2009-14464-C02-02 (Ministerio de Ciencia e Innovación) and Incite09 207 215 PR (Xunta de Galicia). The second author was also partially supported by Plan Nacional de Movilidad de RR.HH. 2008/11, Modalidad “José Castillejo”. 2024-06-21T17:18:12Z 2024-06-21T17:18:12Z 2012 info:eu-repo/semantics/article info:eu-repo/semantics/submittedVersion Rev Mat Complut (2012) 25:109–124 1139-1138 https://uvadoc.uva.es/handle/10324/68187 10.1007/s13163-011-0060-0 109 1 124 Revista Matemática Complutense 25 1988-2807 eng info:eu-repo/semantics/restrictedAccess Revista Matemática Complutense application/pdf Springer https://uvadoc.uva.es/bitstream/10324/68187/3/2012_Corral-Sanz_RevMatCom.pdf.jpg Hispana TEXT http://rightsstatements.org/vocab/CNE/1.0/ UVaDOC. Repositorio Documental de la Universidad de Valladolid https://uvadoc.uva.es/handle/10324/68187