Let $P_1, \cdots, P_n$ be points $\mathbb{P}^2_k$ such that no four of these lie on a line and no seven lie on a conic. Show that the space of plane cubics through these defines a subspace of the space of all plane cubics which can be parametrized by $\mathbb{P}^1_k$
I already know the proof, it is in Reid's undergraduate algebraic geometry p.32. (Also similar things on tao's blog, but a little different.) But I don't like the book's proof, since it contains some cases and corollaries. Can you give me simple proof or any other reference for this?
