Let a matrix $$M=\begin{pmatrix} a_{01}&a_{02}&a_{03}\\a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}$$ with $a_{ij}\in k[x,y,z]$ homogeneous polynomials of degree $d$. Let $p=(\lambda_0: \lambda_1:\lambda_2:\lambda_3)\in\mathbb P^3$ and we set $F_j=\sum\limits_{i=0}^3\lambda_i a_{ij},\ \ \ j=1,2,3$.
Problem:
How many points $p \in\mathbb P^3 $ such that the homogeneous polynomials $F_1,F_2,F_3$ have the greatest common divisor of the positive degree? (It is dependent on the degree $d$.)
Thank you so much!