ON CAMACHO–SAD'S THEOREM ABOUT THE EXISTENCE OF A SEPARATRIX

Abstract

It is proved in Ann. Math. (2)115 (1982) 579–595 that, for any germ of holomorphic nondicritic vector field in (ℂ2, 0), there exists at least one separatrix (invariant analytic curve containing the origin). In Proc. Amer. Math. Soc.125 (1997) 2649–2650 a simple criterion was given to find, at each level of the blow-up, a singular point which leads to an analytical invariant curve. In this paper we prove shortly and strictly combinatorially, the existence of a separatrix, and show that for any germ of holomorphic nondicritic vector field in (ℂ2, 0), there exists at least one separatrix issuing from each connected component of the exceptional divisor of its nice blow-up with nodal corner points deleted.