on connectivity, Zariski connectedness theorem
Let be a proper surjective morphism of irreducible varieties, let the field of rational functions be separably algebraically closed in and let be a normal point; then is connected (moreover, geometrically connected) (see ). 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 is an open imbedding into a neighbourhood of a normal point if is a finite set (see ). In particular, a birational morphism of normal varieties which is bijective at points is an isomorphism. Another formulation of this theorem: Let be a quasi-finite separable morphism of schemes, and let be a quasi-compact quasi-separable scheme; then there exists a decomposition , where is a finite morphism and an open imbedding .
|||O. Zariski, "Foundations of a general theory of birational correspondences" Trans. Amer. Math. Soc. , 53 : 3 (1943) pp. 490–542 MR0008468 Zbl 0061.33004|
|||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|
|[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|
Zariski theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Zariski_theorem&oldid=24018