Editing Branch point (section)

==Algebraic geometry==
{{Main|Branched covering}}
{{See also|Unramified morphism}}
In the context of [[algebraic geometry]], the notion of branch points can be generalized to mappings between arbitrary [[algebraic curve]]s.  Let ƒ:''X''&nbsp;→&nbsp;''Y'' be a morphism of algebraic curves.  By pulling back rational functions on ''Y'' to rational functions on ''X'', ''K''(''X'') is a [[field extension]] of ''K''(''Y'').  The degree of ƒ is defined to be the degree of this field extension [''K''(''X''):''K''(''Y'')], and ƒ is said to be finite if the degree is finite.

Assume that ƒ is finite.  For a point ''P''&nbsp;∈&nbsp;''X'', the ramification index ''e''<sub>''P''</sub> is defined as follows.  Let ''Q''&nbsp;=&nbsp;ƒ(''P'') and let ''t'' be a [[local parameter|local uniformizing parameter]] at ''P''; that is, ''t'' is a regular function defined in a neighborhood of ''Q'' with ''t''(''Q'')&nbsp;=&nbsp;0 whose differential is nonzero.  Pulling back ''t'' by ƒ defines a regular function on ''X''.  Then
:<math>e_P = v_P(t\circ f)</math>
where ''v''<sub>''P''</sub> is the [[valuation ring|valuation]] in the local ring of regular functions at ''P''.  That is, ''e''<sub>''P''</sub> is the order to which <math>t\circ f</math> vanishes at ''P''.  If ''e''<sub>''P''</sub>&nbsp;>&nbsp;1, then ƒ is said to be ramified at ''P''.  In that case, ''Q'' is called a branch point.