Unibranch local ring
In algebraic geometry, a local ring A is said to be unibranch if the reduced ring Ared 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 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 a proper dominant morphism. Denote their Function [field (scheme theory)|function fields] by K and K, respectively. Suppose that the algebraic closure of K in K has separable degree n and that is unibranch. Then the fiber 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.