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 **topologically** 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.

I tried to reproduce this picture on a computer, using the map $\mathbb{CP}^2\to\mathbb{R}^7$ given by

$(z_1:z_2:z_3)\mapsto(z_2\overline{z_3},z_3\overline{z_1},z_1\overline{z_2},|z_1|^2-|z_2|^2)/(|z_1|^2+|z_2|^2+|z_3|^2)$,

projecting to various $\mathbb{R}^3$s, and looking for $L$ by trial and error; all in vain. Which brings me to....

Questions:

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) ?

Is there an algorithm to find such a line ?

Is there a "better" way to project an open part of the complex projective plane to $\mathbb{R}^3$ ?