# Branch index

From Encyclopedia of Mathematics

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.

#### References

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

#### Comments

#### References

[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: http://encyclopediaofmath.org/index.php?title=Branch_index&oldid=46147

This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article