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Jun 13, 2018 at 17:25 comment added Dima Pasechnik OK --- indeed, all you need is that the space of real cubics vanishing on the 8 points is 2-dimensional, and $C_1$ and $C_2$ do not share a component.
Jun 13, 2018 at 17:23 vote accept Dima Pasechnik
Jun 13, 2018 at 16:37 comment added Dima Pasechnik Well, this is the intuition (one can probably even replace "random" by "no 4 on a line, no 7 on a conic"). But I don't know how one does make this into a proof.
Jun 13, 2018 at 16:29 history answered Noam D. Elkies CC BY-SA 4.0