Invariant curves for holomorphic foliations on singular surfaces (Q2180950)

The Separatrix Theorem of \textit{C. Camacho} and \textit{P. Sad} [Ann. Math. (2) 115, 579--595 (1982; Zbl 0503.32007)] says that there exists at least one invariant curve (separatrix) passing through the singularity of a germ of holomorphic foliation on a 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. For the singular case see [\textit{M. Sebastiani}, An. Acad. Bras. Ciênc. 69, No. 2, 159--162 (1997; Zbl 0887.57033)]. The author proves the existence of separatrix even when the resolution dual graph of the surface singular point is not a tree. The main result is as follows. Let \(\mathcal{F}\) be a singular holomorphic foliation on a normal singular surface \(X\). If the foliation has no saddle-node in its resolution/reduction over the singularity \(p \in X\) and the normal sheaf \(N_F\) is \(\mathbb{Q}\)-Gorenstein, then \(F\) has a separatrix through \(p\).