Talk:Hyperelliptic curve

Quartic image
Isn't the image not a hyperelliptic curve since f(x) is quartic in this case? Akriasas (talk) 20:53, 11 December 2007 (UTC)


 * Absolutely. This page needs a total rewrite. The stuff about singularities at infinity is incorrect as well. I don't have time to rewrite now though, but maybe later. :) 85.225.78.253 (talk) 09:57, 29 March 2008 (UTC)

Definition
A particularly short definition of a hyperelliptic curve can be given as follows. Say C = complex numbers. A curve over C is hyperelliptic iff its function field has a unique index-2 subfield that is purely transcendental over C. (Mark van Hoeij, Oct 12, 2011). — Preceding unsigned comment added by 128.186.104.253 (talk) 19:32, 12 October 2011 (UTC)

Image
I replaced the image with an image of a hyperelliptic curve — Preceding unsigned comment added by Tbonefrog (talk • contribs) 11:14, 7 January 2013 (UTC)

Bijective?
The article says:


 * Suppose the bijective morphism f : X → P1 with ramification degree 2, where X is a curve with genus g and P1 is the Riemann sphere.

Am I confused? How can f be a bijection from a hyperelliptic curve to the Riemann sphere except in the very special case where X *is* the Riemann sphere? I think we should delete "bijective morphism" and say something like "branched covering". John Baez (talk) 04:08, 12 December 2021 (UTC)