On one hand the real locus of a complex elliptic curve is the intersection of a plane with a torus (i.e. a torus embedded in $\mathbb{C}^2$ plus infinity). And an elliptic curve has no cusps or self-intersections. But the intersection of a plane with a non-singular torus can have those.


I suppose those intersections of a complex cubic curve with a plane which can (up to some complex projective change of coordinates) give a real locus cannot have cusps or self intersections unless the cubic does too. 


But is that right? Is it easy to see, once you see it? If it is not right what is? 



Visualizing singular points of real loci of elliptic curves
The tangent plane to an elliptic curve at a point is a holomorphic curve, meaning it is closed under multiplication by $i$. The real plane does not intersect its product with $i$, Which is the imaginary plane. Hence they cannot be identical, which would happen at a cusp or self-intersection point.


