# Zariski theorem

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 [2]). 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 [1]). 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 .

#### References

 [1] O. Zariski, "Foundations of a general theory of birational correspondences" Trans. Amer. Math. Soc. , 53 : 3 (1943) pp. 490–542 MR0008468 Zbl 0061.33004 [2] O. Zariski, "Theory and applications of holomorphic functions on algebraic varieties over arbitrary ground fields" Mem. Amer. Math. Soc. , 5 (1951) pp. 1–90 MR0041487 [3a] A. Grothendieck, "Eléments de géometrie algébrique. III. Etude cohomologique des faisceaux cohérents I" Publ. Math. IHES , 11 (1961) MR0217085 MR0163910 [3b] A. Grothendieck, "Eléments de géometrie algébrique. IV. Etude locale des schémas et des morphismes des schémas IV" Publ. Math. IHES , 32 (1967) MR0238860 Zbl 0144.19904 Zbl 0135.39701 Zbl 0136.15901