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 géométrie algébrique. III. Etude cohomologique des faisceaux cohérents I" Publ. Math. IHES , 11 (1961) MR0217085 MR0163910
[3b]	A. Grothendieck, "Eléments de géométrie 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

Comments

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]).

References

[a1]	R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001
[a2]	O. Zariski, "The connectedness theorem for birational transformations" R.H. Fox (ed.) D.C. Spencer (ed.) A.W. Tucker (ed.) , Algebraic geometry and topology (Symp. in honor of S. Lefschetz) , Princeton Univ. Press (1957) pp. 182–188 MR0090099 Zbl 0087.35601
[a3]	J.P. Murre, "On a connectedness theorem for a birational transformation at a simple point" Amer. J. Math. , 80 (1958) pp. 3–15 MR0093524 Zbl 0087.35602
[a4]	W.-L. Chow, "On the connectedness theorem in algebraic geometry" Amer. J. Math. , 83 (1959) pp. 1033–1074 MR0110705 Zbl 0192.26806