stability of linear systems of quadrics

Question

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?

Answer 1

This is a partial answer.

There are many papers on nets of quadrics (see e.g. work of Debarre and Beauville). Here the discriminant locus is a plane curve. If the base-locus of the net is non-singular, then this curve is stable, but need not be non-singular in general (e.g. singular discriminant curves arise from a net generated by sufficiently general diagonal quadrics). I don't know whether the base-locus being stable implies that the discriminant locus is stable, or conversely. People are certainly interested in the case where the base-locus is singular however, again I would recommend the papers by the above authors as a good starting place.

For higher-dimensional linear systems of quadrics, things get increasingly more complicated and much less is known. I would be surprised if the results you wanted were known in such cases. Certainly the case of nets is the best place to look first.

Answer 2

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.

In particular, they show that the (semi-)stability of a net cannot be read off the (semi-)stability of the disriminant locus (see for instance Remark 3.21).

[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.]