algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, Zariski's connectedness theorem (due to

Oscar Zariski , birth_date = , birth_place = Kobrin, Russian Empire , death_date = , death_place = Brookline, Massachusetts, United States , nationality = American , field = Mathematics , work_institutions = ...

) says that under certain conditions the fibers of a morphism of varieties are connected. It is an extension of

Zariski's main theorem In algebraic geometry, Zariski's main theorem, proved by , is a statement about the structure of birational morphisms stating roughly that there is only one branch at any normal point of a variety. It is the special case of Zariski's connectedness ...

to the case when the morphism of varieties need not be birational. Zariski's connectedness theorem gives a rigorous version of the "principle of degeneration" introduced by

Federigo Enriques Abramo Giulio Umberto Federigo Enriques (5 January 1871 – 14 June 1946) was an Italian mathematician, now known principally as the first to give a classification of algebraic surfaces in birational geometry, and other contributions in algebrai ...

, which says roughly that a limit of absolutely irreducible cycles is absolutely connected.

Statement

Suppose that ''f'' is a

proper Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...

surjective

morphism of varieties In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular ...

from ''X'' to ''Y'' such that the function field of ''Y'' is

separably closed In field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial of \alpha over is a separable polynomial (i.e., its formal derivative is not the zero polynom ...

in that of ''X''. Then Zariski's connectedness theorem says that the inverse image of any normal point of ''Y'' is connected. An alternative version says that if ''f'' is proper and ''f''_* ''O''_''X'' = ''O''_''Y'', then ''f'' is surjective and the inverse image of any point of ''Y'' is connected.

References

* *{{citation, mr=0090099, last=Zariski, first= Oscar, chapter=The connectedness theorem for birational transformations, title= Algebraic geometry and topology. A symposium in honor of S. Lefschetz, pages= 182–188, publisher= Princeton University Press, place= Princeton, N. J., year= 1957 Theorems in algebraic geometry