polynomials satisfying the Plücker relation

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
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.


Source : Link , Question Author : Hephaistos , Answer Author : Community

Leave a Comment