Arkiv för Matematik

Volume 58 (2020)

Number 1

Invariant curves for holomorphic foliations on singular surfaces

Pages: 179 – 195



Edileno de Almeida Santos (Instituto de Ciência, Engenharia e Tecnologia, Universidade Federal dos Vales do Jequitinhonha e Mucuri (UFVJM), Teófilo Otoni, MG, Brazil)


The Separatrix Theorem of C. Camacho and P. Sad says that there exists at least one invariant curve (separatrix) passing through the singularity of a germ of holomorphic foliation on complex surface, when the surface underlying the foliation is smooth or when it is singular and the dual graph of resolution surface singularity is a tree. Under some assumptions, we obtain existence of separatrix even when the resolution dual graph of the surface singular point is not a tree. It will be necessary to require an extra condition of the foliation, namely, absence of saddle-node in its reduction of singularities.


foliations, invariant curves, birational geometry

2010 Mathematics Subject Classification


Received 7 November 2018

Received revised 10 September 2019

Accepted 27 September 2019

Published 21 July 2022