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:

Several results in commutative algebra imply the geometric forms of Zariski's main theorem, including:

The original result was labelled as the "MAIN THEOREM" in Zariski (1943).

Let f be a birational mapping of algebraic varieties V and W. Recall that f is defined by a closed subvariety ${\displaystyle \Gamma \subset V\times W}$ (a "graph" of f) such that the projection on the first factor ${\displaystyle p_{1}}$ induces an isomorphism between an open ${\displaystyle U\subset V}$ and ${\displaystyle p_{1}^{-1}(U)}$, and such that ${\displaystyle p_{2}\circ p_{1}^{-1}}$ 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 ${\displaystyle p_{2}\circ p_{1}^{-1}}$ is called a total transform of it.

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. Hartshorne (1977, Corollary III.11.4) calls the following connectedness statement "Zariski's Main theorem":