Visualizing a complex plane cubic together with the real plane - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T13:22:12Z http://mathoverflow.net/feeds/question/20007 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/20007/visualizing-a-complex-plane-cubic-together-with-the-real-plane Visualizing a complex plane cubic together with the real plane David Lehavi 2010-03-31T21:34:29Z 2010-05-03T19:36:48Z <p>In Alain Roberts "Elliptic curves: notes from postgraduate lectures given in Lausanne 1971/72" page 11 (available on google books unless you already tried to read another chapter), there is a hand drawn picture of a real 2-dimensional torus and a real plane, which <strong>topologically</strong> represent the way a complex cubic (with two real components) and the real projective plane sit in the complex projective plane. Taking the picture on face value, one should be able to project an open subset of the complex projective plane to $\mathbb{R}^3$, so that there is some real line $L$ that passes through the "doughnut" defined by the image of the complex cubic.</p> <p>I tried to reproduce this picture on a computer, using the map $\mathbb{CP}^2\to\mathbb{R}^7$ given by</p> <p>$(z_1:z_2:z_3)\mapsto(z_2\overline{z_3},z_3\overline{z_1},z_1\overline{z_2},|z_1|^2-|z_2|^2)/(|z_1|^2+|z_2|^2+|z_3|^2)$,</p> <p>projecting to various $\mathbb{R}^3$s, and looking for $L$ by trial and error; all in vain. Which brings me to....</p> <p>Questions:</p> <ul> <li><p>Is there such a line (the map I used does not send the real projective plane to a plane, so it does not have to be the case even if Roberts picture is correct) ?</p></li> <li><p>Is there an algorithm to find such a line ? </p></li> <li><p>Is there a "better" way to project an open part of the complex projective plane to $\mathbb{R}^3$ ?</p></li> </ul> http://mathoverflow.net/questions/20007/visualizing-a-complex-plane-cubic-together-with-the-real-plane/23367#23367 Answer by Kristal Cantwell for Visualizing a complex plane cubic together with the real plane Kristal Cantwell 2010-05-03T19:36:48Z 2010-05-03T19:36:48Z <p>I found this article: "Visualizing Elliptic Curves" by Donu Arapura it is available at the following URL: <a href="http://www.math.purdue.edu/~dvb/graph/elliptic.pdf" rel="nofollow">http://www.math.purdue.edu/~dvb/graph/elliptic.pdf</a> In it he discusses a projection that sends sends the real part of $x$ to $x_1$ and the real part of $y$ to $x_3$ thus it would seem to preserve the entire real plane and any line in it. So this might be useful to you.</p>