Our aim is to understand the algebraic notion of flatness in explicit geometric terms. Let be a morphism of complex-analytic spaces, where is smooth. We prove that nonflatness of is equivalent to a severe discontinuity of the fibres---the existence of a {\it vertical component} (a local irreducible component at a point of the source whose image is nowhere-dense in )---after passage to the -fold fibred power of , where . Our main theorem is a more general criterion for flatness over of a coherent sheaf of modules on . In the case that is a morphism of complex algebraic varieties, the result implies that the stalk of at a point is flat over if and only if its -fold tensor power is a torsion-free -module (conjecture of Vasconcelos in the case of -algebras).