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.
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