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UDIBRANCH

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Udibranch might refer to
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. The notions of unibranch and geometrically unibranch points are used in some theorems in algebraic geometry. For example, there is the following result: * Theorem (EGA, III.4.3.7) 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.

Udibranch might refer to

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. The notions of unibranch and geometrically unibranch points are used in some theorems in algebraic geometry. For example, there is the following result:
* Theorem (EGA, III.4.3.7) 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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