Zariski's Connectedness Theorem

In algebraic geometry, Zariski's connectedness theorem (due to Oscar Zariski) says that under certain conditions the fibers of a morphism of varieties are connected. It is an extension of Zariski's main theorem to the case when the morphism of varieties need not be birational.

Zariski's connectedness theorem gives a rigorous version of the "principle of degeneration" introduced by Federigo Enriques, which says roughly that a limit of absolutely irreducible cycles is absolutely connected.

Statement

Suppose that ''f'' is a proper surjective morphism of varieties from ''X'' to ''Y'' such that the function field of ''Y'' is separably closed in that of ''X''. Then Zariski's connectedness theorem says that the inverse image of any normal point of ''Y'' is connected. An alternative version says that if ''f'' is proper and ''f''_* ''O''_''X'' = ''O''_''Y'', then ''f'' is surjective and the inverse image of any point of ''Y'' is connected.

References

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*{{citation, mr=0090099, last=Zariski, first= Oscar, chapter=The connectedness theorem for birational transformations, title= Algebraic geometry and topology. A symposium in honor of S. Lefschetz, pages= 182–188, publisher= Princeton University Press, place= Princeton, N. J., year= 1957

Theorems in algebraic geometry