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

• Zariski, Oscar (1951), Theory and applications of holomorphic functions on algebraic varieties over arbitrary ground fields, Memoirs of the American Mathematical Society, vol. 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