Let E in P3be a real elliptic normal curve with two non-null-homotopic components. Is there a parametrization (R/Z) x (Z/2Z) -> E such that any four points on E are coplanar precisely when their corresponding parameters sum to zero? As an example one could take E to be the complete intersection of the quadrics XY + ZW = 0 and -X2 + Y2 - 2Z2 + ZW + 2W2 = 0.
For more details (and better formatting), see the corresponding question on MathOverflow: http://mathoverflow.net/questions/197848/ .
http://mathoverflow.net/questions/197848/
An answer to this question was provided by Alex Degtyarev on MathOverflow. To paraphrase: If such a parametrization exists, the image of zero would have to be a hyperflex point on one of the real connected components, in which the curve has 4-fold intersection with a plane. Identify the complement of this plane with R3. Since the curve has degree four, this plane cannot intersect the other component, which is therefore contained in R3. This other component is therefore null-homologous, contradicting our initial assumption.
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