Zariski's main theorem

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:

The name "Zariski's main theorem" comes from the fact that Zariski labelled it 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 indeteminancy locus, and an image of a subset of V under ${\displaystyle p_{2}\circ p_{1}^{-1}}$ is called a total transform of it.

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