# stability of linear systems of quadrics

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, "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?

The paper "Stability of genus five canonical curves " by Fedorchuk and Smyth (arXiv:1302.4804) contains a detailed analysis of the (semi-)stability of nets of quadrics in $\mathbb{P}^4$, and completely describes the GIT quotient.
[Fedorchuk and Smyth's motivation is to construct a birational model of $\overline{M}_5$, and to give a modular interpretation of it. This work is part of the general idea to try to run explicitely the MMP for moduli spaces of curves.]