Let $S_{12}$, $S_{13}$, $S_{14}$, $S_{23}$, $S_{24}$, $S_{34}$ be complex homogeneous polynomials in 4 variables satisfying the Plücker relation : $$S_{12}S_{34}-S_{13}S_{24}+S_{14}S_{23}=0 .$$ Suppose also that they don't have any common non trivial zero. Let $d_{ij}=\deg(S_{ij})$. Let $d$ be the integer $$d=d_{12}+d_{34}=d_{13}+d_{24}=d_{14}+d_{23}.$$ By an indirect way (I used this to construct vector bundles on $\mathbb{P}_3$), I find that $d$ should be a divisor of $d_{12}d_{13}d_{14}$. I would like to know if this is correct, and if it is true what is a direct proof.

is there a Pythagorean 6-tuple of odd degree complex homogeneous polynomials in 4 variables without nontrivial common zero? Forrealpolynomials the latter question is answered easily. Cf. mathoverflow.net/questions/62820/pythagorean-5-tuples. $\endgroup$