Talk:Zariski's connectedness theorem

Discussion of accuracy
The first version in the "Statement" section is false: consider a finite map of smooth projective curves over an algebraically closed field. It is proper and surjective, the extension of function fields is automatically separably generated, yet almost all fibers are disconnected (unless the map is an isomorphism).

The second version is fine. The above map does not satisfy the hypotheses if it is not an isomorphism.

The first version becomes true if we require the map to be birational, but this statement already appears in the article on Zariski's main theorem. (as the second sentence currently includes a requirement for the map to be birational) Klbrain (talk) 10:42, 5 November 2017 (UTC)