compact riemann surface of genus g - MathOverflow 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 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 Answer by S. Carnahan for compact riemann surface of genus g 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>