# Zariski theorem

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on connectivity, Zariski connectedness theorem

Let be a proper surjective morphism of irreducible varieties, let the field of rational functions be separably algebraically closed in and let be a normal point; then is connected (moreover, geometrically connected) (see ). The theorem provides a basis for the classical principle of degeneration: If the generic cycle of an algebraic system of cycles is a variety (i.e. is geometrically irreducible), then any specialization of that cycle is connected.

A special case of the Zariski connectedness theorem is the so-called fundamental theorem of Zariski, or Zariski's birational correspondence theorem: A birational morphism of algebraic varieties is an open imbedding into a neighbourhood of a normal point if is a finite set (see ). In particular, a birational morphism of normal varieties which is bijective at points is an isomorphism. Another formulation of this theorem: Let be a quasi-finite separable morphism of schemes, and let be a quasi-compact quasi-separable scheme; then there exists a decomposition , where is a finite morphism and an open imbedding .