View source for Zariski theorem

''on connectivity, Zariski connectedness theorem''

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099130/z0991301.png" /> be a proper surjective [[Morphism|morphism]] of irreducible varieties, let the field of rational functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099130/z0991302.png" /> be separably algebraically closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099130/z0991303.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099130/z0991304.png" /> be a normal point; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099130/z0991305.png" /> is connected (moreover, geometrically connected) (see [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099130/z0991306.png" /> is an open imbedding into a neighbourhood of a normal point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099130/z0991307.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099130/z0991308.png" /> is a finite set (see [[#References|[1]]]). In particular, a birational morphism of normal varieties which is bijective at points is an isomorphism. Another formulation of this theorem: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099130/z0991309.png" /> be a quasi-finite separable morphism of schemes, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099130/z09913010.png" /> be a quasi-compact quasi-separable scheme; then there exists a decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099130/z09913011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099130/z09913012.png" /> is a finite morphism and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099130/z09913013.png" /> an open imbedding .

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> O. Zariski, "Foundations of a general theory of birational correspondences" ''Trans. Amer. Math. Soc.'' , '''53''' : 3 (1943) pp. 490–542 {{MR|0008468}} {{ZBL|0061.33004}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> O. Zariski, "Theory and applications of holomorphic functions on algebraic varieties over arbitrary ground fields" ''Mem. Amer. Math. Soc.'' , '''5''' (1951) pp. 1–90 {{MR|0041487}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top"> A. Grothendieck, "Eléments de géometrie algébrique. III. Etude cohomologique des faisceaux cohérents I" ''Publ. Math. IHES'' , '''11''' (1961) {{MR|0217085}} {{MR|0163910}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top"> A. Grothendieck, "Eléments de géometrie algébrique. IV. Etude locale des schémas et des morphismes des schémas IV" ''Publ. Math. IHES'' , '''32''' (1967) {{MR|0238860}} {{ZBL|0144.19904}} {{ZBL|0135.39701}} {{ZBL|0136.15901}} </TD></TR></table>



====Comments====
In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099130/z09913014.png" /> is a proper birational morphism and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099130/z09913015.png" /> is a non-singular point, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099130/z09913016.png" /> is moreover linearly connected, i.e. any two points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099130/z09913017.png" /> can be connected by a sequence of rational curves in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099130/z09913018.png" /> (see [[#References|[a2]]]–[[#References|[a4]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> 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 {{MR|0090099}} {{ZBL|0087.35601}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J.P. Murre, "On a connectedness theorem for a birational transformation at a simple point" ''Amer. J. Math.'' , '''80''' (1958) pp. 3–15 {{MR|0093524}} {{ZBL|0087.35602}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> W.-L. Chow, "On the connectedness theorem in algebraic geometry" ''Amer. J. Math.'' , '''83''' (1959) pp. 1033–1074 {{MR|0110705}} {{ZBL|0192.26806}} </TD></TR></table>