Zariski's Main Theorem - Zariski's Main Theorem For Birational Morphisms

Zariski's Main Theorem For Birational Morphisms

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 indeteminancy locus, and an image of a subset of V under is called a total transform of it.

The original statement of the theorem in (Zariski 1943, p. 522) 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 is of higher dimension than W.

Here are some variants of this theorem stated using modern terminology. Hartshorne (1977) calls the following connectedness statement (Corollary III.11.4,loc.cit):

If f:X→Y is a birational projective morphism between noetherian integral schemes, then the inverse image of every normal point of Y is connected.

a "Zariski's Main theorem". The following consequence of it (Theorem V.5.2,loc.cit.) also goes under this name:

If f:X→Y is a birational transformation of projective varieties with Y normal, then the total transform of a fundamental point of f is connected and of dimension at least 1.