Zariski theorem

on connectivity, Zariski connectedness theorem

Let $f : X \rightarrow Y$ be a proper surjective morphism of irreducible varieties, let the field of rational functions $k ( Y )$ be separably algebraically closed in $k ( X )$ and let $y \in Y$ be a normal point; then $f ^ { - 1 } ( y )$ is connected (moreover, geometrically connected) (see [2]). The theorem provides a basis for the classical principle of degeneration: If the generic cycle of an algebraic system of cycles is a variety (i.e. is geometrically irreducible), then any specialization of that cycle is connected.

A special case of the Zariski connectedness theorem is the so-called fundamental theorem of Zariski, or Zariski's birational correspondence theorem: A birational morphism of algebraic varieties $f : X \rightarrow Y$ is an open imbedding into a neighbourhood of a normal point $y \in Y$ if $f ^ { - 1 } ( y )$ is a finite set (see [1]). In particular, a birational morphism of normal varieties which is bijective at points is an isomorphism. Another formulation of this theorem: Let $f : X \rightarrow Y$ be a quasi-finite separable morphism of schemes, and let $Y$ be a quasi-compact quasi-separable scheme; then there exists a decomposition $f = u \circ g$, where $u$ is a finite morphism and $g$ an open imbedding .

References

 [1] O. Zariski, "Foundations of a general theory of birational correspondences" Trans. Amer. Math. Soc. , 53 : 3 (1943) pp. 490–542 MR0008468 Zbl 0061.33004 [2] O. Zariski, "Theory and applications of holomorphic functions on algebraic varieties over arbitrary ground fields" Mem. Amer. Math. Soc. , 5 (1951) pp. 1–90 MR0041487 [3a] A. Grothendieck, "Eléments de géometrie algébrique. III. Etude cohomologique des faisceaux cohérents I" Publ. Math. IHES , 11 (1961) MR0217085 MR0163910 [3b] A. Grothendieck, "Eléments de géometrie algébrique. IV. Etude locale des schémas et des morphismes des schémas IV" Publ. Math. IHES , 32 (1967) MR0238860 Zbl 0144.19904 Zbl 0135.39701 Zbl 0136.15901

In case $f: X \rightarrow Y$ is a proper birational morphism and $y \in Y$ is a non-singular point, $f ^ { - 1 } ( y)$ is moreover linearly connected, i.e. any two points of $f ^ { - 1 } ( y)$ can be connected by a sequence of rational curves in $f ^ { - 1 } ( y)$( see [a2][a4]).