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Aug 17, 2010 at 8:09 comment added MartinG No genericity condition is needed: any line intersects the cubic in an effective degree 3 divisor summing to zero, and conversely. So the result is isomorphic to the projective plane (it is also the linear system |0+0+0|, where 0 means the group identity).
Aug 17, 2010 at 0:43 history answered Felipe Voloch CC BY-SA 2.5