most recent 30 from http://mathoverflow.net 2013-05-25T05:54:12Z http://mathoverflow.net/feeds/question/23508 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23508/compact-riemann-surface-of-genus-g priyanka 2010-05-04T23:07:53Z 2010-05-05T00:26:06Z

<p>Is there a way to construct a compact Riemann Surface X of genus[topological] g when g is given.</p>

http://mathoverflow.net/questions/23508/compact-riemann-surface-of-genus-g/23520#23520 S. Carnahan 2010-05-05T00:26:06Z 2010-05-05T00:26:06Z

<p>Choose $2g+2$ distinct complex numbers $z_i$, and take a double cover of $\mathbb{P}^1$ branched at these points. This is typically written as a plane curve with an affine patch defined by $y^2 = \prod (x-z_i)$. See <a href="http://en.wikipedia.org/wiki/Hyperelliptic_curve" rel="nofollow">the Wikipedia article</a> (which could use some polish).</p> <p>Topologically, you can picture a genus $g$ surface with the handles lined up in a row, and take a quotient by a 180 degree rotation along the axis of symmetry to get a sphere.</p>