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I do not know why $M(K_4)$ is not affine over $GF(2)$ or $GF(3)$ but it is affine over all fields with more than 3 elements. I proved that $U_{2,4}$ is $\mathbb F$-representable iff $|\mathbb F| \geq 3.$

Is there any coincidence between the numbers that help in concluding why $M(K_4)$ is not affine over $GF(2)$ or $GF(3)$ but it is affine over all fields with more than 3 elements?

Any explanation will be greatly appreciated!

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  • $\begingroup$ Are you familiar with the fact that $M$ is $\mathbb{F}$-representable iff it is a restriction of $PG(r(M)-1, \mathbb{F})$? $\endgroup$
    – Mathieu Rundström
    Commented Jan 7 at 22:35
  • $\begingroup$ Yes I am @MathieuRundström $\endgroup$
    – Intuition
    Commented Jan 7 at 23:01
  • $\begingroup$ but I still do not see what you are trying to tell me @MathieuRundström $\endgroup$
    – Intuition
    Commented Jan 8 at 2:03

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