View source for Branch index

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The sum  $  V= \sum (k - 1) $
of the orders of the branch points (cf. [[Branch point|Branch point]]) of a compact Riemann surface  $  S $, 
regarded as an  $  n $-
sheeted covering surface over the Riemann sphere, extended over all finite and infinitely-distant branch points of  $  S $. 
The branch index is connected with the genus  $  g $
and number of sheets  $  n $
of  $  S $
by:

$$ 
V  =  2 (n + g - 1).
$$

See also [[Riemann surface|Riemann surface]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10 {{MR|0092855}} {{ZBL|0078.06602}} </TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D. Mumford, "Algebraic geometry" , '''1. Complex projective varieties''' , Springer (1976) {{MR|0453732}} {{ZBL|0356.14002}} </TD></TR></table>