Difference between revisions of "Branch index"
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| + | $#C+1 = 8 : ~/encyclopedia/old_files/data/B017/B.0107480 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]]. | See also [[Riemann surface|Riemann surface]]. | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <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> |
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====Comments==== | ====Comments==== | ||
| − | |||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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> |
Latest revision as of 06:29, 30 May 2020
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=13651
Branch index. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Branch_index&oldid=13651
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article