Vertical components in fibre powers of analytic spaces

We study the relationship between degeneracies of the family of fibres of an analytic mapping and the existence of vertical components in fibre powers of the mapping. Our main result is the following criterion for openness of complex analytic maps: Let f :X→Y be an analytic map of analytic spaces, with X being puredimensional and Y being locally irreducible of dimension n. Then f is open if and only if there are no isolated algebraic vertical components in the nth fibre power X{n}.