Zariski's connectedness theorem

In algebraic geometry, Zariski's connectedness theorem 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 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

Zariski, Oscar (1951), Theory and applications of holomorphic functions on algebraic varieties over arbitrary ground fields, Mem. Amer. Math. Soc., 5, MR 0041487
Zariski, Oscar (1957), "The connectedness theorem for birational transformations", Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton, N. J.: Princeton University Press, pp. 182–188, MR 0090099

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