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
* *{{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