Theorem of algebraic geometry and commutative algebra
In algebraic geometry, Zariski's main theorem, proved by Oscar Zariski (1943), is a statement about the structure of birational morphisms stating roughly that there is only one branch at any normal point of a variety. It is the special case of Zariski's connectedness theorem when the two varieties are birational.
Zariski's main theorem can be stated in several ways which at first sight seem to be quite different, but are in fact deeply related. Some of the variations that have been called Zariski's main theorem are as follows:
- A birational morphism with finite fibers to a normal variety is an isomorphism to an open subset.
- The total transform of a normal fundamental point of a birational map has positive dimension. This is essentially Zariski's original version.
- The total transform of a normal point under a proper birational morphism is connected.
- A generalization due to Grothendieck describes the structure of quasi-finite morphisms of schemes.
Several results in commutative algebra imply the geometric forms of Zariski's main theorem, including:
- A normal local ring is unibranch, which is a variation of the statement that the transform of a normal point is connected.
- The local ring of a normal point of a variety is analytically normal. This is a strong form of the statement that it is unibranch.
The original result was labelled as the "MAIN THEOREM" in Zariski (1943).