Branch index

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The sum $ V= \sum (k - 1) $ of the orders of the branch points (cf. 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.


[1] G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10 MR0092855 Zbl 0078.06602



[a1] D. Mumford, "Algebraic geometry" , 1. Complex projective varieties , Springer (1976) MR0453732 Zbl 0356.14002
How to Cite This Entry:
Branch index. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article