Concept
In algebraic geometry, Zariski's main theorem, proved by , 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: The total transform of a normal fundamental point of a birational map has positive dimension. This is essentially Zariski's original form of his main theorem. A birational morphism with finite fibers to a normal variety is an isomorphism to an open subset. The total transform of a normal point under a proper birational morphism is connected. A closely related theorem of Grothendieck describes the structure of quasi-finite morphisms of schemes, which implies Zariski's original main theorem. Several results in commutative algebra that imply the geometric form of Zariski's main theorem. 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 name "Zariski's main theorem" comes from the fact that Zariski labelled it as the "MAIN THEOREM" in . Let f be a birational mapping of algebraic varieties V and W. Recall that f is defined by a closed subvariety (a "graph" of f) such that the projection on the first factor induces an isomorphism between an open and , and such that is an isomorphism on U too. The complement of U in V is called a fundamental variety or indeterminacy locus, and the image of a subset of V under is called a total transform of it. The original statement of the theorem in reads: MAIN THEOREM: If W is an irreducible fundamental variety on V of a birational correspondence T between V and V′ and if T has no fundamental elements on V′ then — under the assumption that V is locally normal at W — each irreducible component of the transform T[W] is of higher dimension than W.