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.
Let f be a birational mapping of algebraic varieties V and W. Recall that f is defined by a closed subvariety
The complement of U in V is called a fundamental variety or indeterminacy locus, and the image of a subset of V under
The original statement of the theorem in (Zariski 1943, p. 522) reads: Here T is essentially a morphism from V′ to V that is birational, W is a subvariety of the set where the inverse of T is not defined whose local ring is normal, and the transform T[W] means the inverse image of W under the morphism from V′ to V. Here are some variants of this theorem stated using more recent terminology.
Then structure theorem for quasi-finite morphisms applies and yields the desired result.
Zariski (1949) reformulated his main theorem in terms of commutative algebra as a statement about local rings.
This is essentially Zariski's formulation of his main theorem in terms of commutative rings.
A topological version of Zariski's main theorem says that if x is a (closed) point of a normal complex variety it is unibranch; in other words there are arbitrarily small neighborhoods U of x such that the set of non-singular points of U is connected (Mumford 1999, III.9).