A pencil of quadrics in $\mathbb{P}^n$ is a line in $\mathbb{P}^N$, where $N=\frac{n(n+3)}{2}$. So the space of pencil of quadrics is the Grassmannian $Gr(2,N+1)$. The group $SL_{n+1}(\mathbb{C})$ acts on $Gr(2,N+1)$ by $T\circ (a_1A+a_2B)=a_1T^tAT+a_2T^tBT$ for any $T\in SL_{n+1}(\mathbb{C})$ and pencil $a_1A+a_2B$.(Here, $A,B$: symmetric $(n+1)\times (n+1)$ matrices corresponding to quadrics.) Thinking $Gr(2,N+1)$ as embedded into projective space via $Pl\ddot{u}cker$ mapping we have the notion of stability of GIT for pencils of quadrics.

And $D(a_1,a_2):=det(a_1A+a_2B)$ is a polynomial of degree $n+1$ in the two variables. $D(a_1,a_2)$ has roots in $\mathbb{P}^1$. There is natural $SL_{2}(\mathbb{C})$ action. 

From [this paper](http://www.mat.ufmg.br/~dan/artigos/pencils_of_quadric.pdf), "stability of pencils of quadrics in $\mathbb{P}^n$"="stability of $n+1$ points in $\mathbb{P}^1$." 

I want to know I can tell same(or similar) thing about "stability of linear systems(net, web or any linear system) of quadrics". Can You give any results or references?