In algebraic geometry, Zariski's main theorem, proved by Oscar Zariski, states that there is only one branch at any point of a normal variety. Informally, the reason it is true is that any branch locus is a singularity of codimension 1, while singularities of normal varieties all have codimension at least 2.
Zariski's main theorem can be stated in several ways which at first sight seem to be quite different, as there are several ways to make the informal notion of having only one branch precise. In particular the name "Zariski's main theorem" is also used for a closely related theorem of Grothendieck that describes the structure of quasi-finite morphisms of schemes, which implies Zariski's original main theorem.
The name "Zariski's main theorem" comes from the fact that it was labeled as the "MAIN THEOREM" in Zariski (1943).
Read more about Zariski's Main Theorem: Zariski's Main Theorem For Birational Morphisms, Zariski's Main Theorem For Quasifinite Morphisms, Zariski's Main Theorem For Commutative Rings, Zariski's Main Theorem: Topological Form, Zariski's Main Theorem: Power Series Form
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