Unibranch local ring

In algebraic geometry, a local ring A is said to be unibranch if the reduced ring Ared (obtained by quotienting A by its nilradical) is an integral domain, and the integral closure B of Ared is also a local ring. A unibranch local ring is said to be geometrically unibranch if the residue field of B is a purely inseparable extension of the residue field of Ared. A complex variety X is called topologically unibranch at a point x if for all complements Y of closed algebraic subsets of X there is a fundamental system of neighborhoods (in the classical topology) of x whose intersection with Y is connected.

In particular, a normal ring is unibranch. One result on unibranch points in algebraic geometry is the following:

Theorem Let X and Y be two integral locally noetherian schemes and

f

:

X

→

Y

{\displaystyle f\colon X\to Y}

a proper dominant morphism. Denote their function fields by K(X) and K(Y), respectively. Suppose that the algebraic closure of K(Y) in K(X) has separable degree n and that

y

∈

Y

{\displaystyle y\in Y}

is unibranch. Then the fiber

f

−

1

(

y

)

{\displaystyle f^{-1}(y)}

has at most n connected components. In particular, if f is birational, then the fibers of unibranch points are connected.

In EGA, the theorem is obtained as a corollary of Zariski's main theorem.