Question regarding semistability of a point of GIT quotient

Question

$\DeclareMathOperator\SL{SL}$I am currently looking at the paper titled "$\SL(2,\mathbb{C})$ quotients de $(\mathbb{P^1})^n$" by Marzia Polito. The author has considered diagonal action of $\SL(2,\mathbb{C})$ over $(\mathbb{P^1})^n$ given by $(g,(x_1, x_2,\cdots,x_n))=(gx_1, gx_2,\cdots,gx_n)$ for $g\in \SL(2,\mathbb{C})$ and $(x_1, x_2,\cdots,x_n)\in (\mathbb{P^1})^n $. The author has considered linearization with respect to the ample line bundle $L(m)=O(m_1)\otimes O(m_2)\cdots O(m_n)$ where $m=(m_1,m_2,\cdots m_n)$ and $|m|=\sum_1^nm_i$ and also defined for $x\in (\mathbb{P^1})^n$ the set $I_k(x)=\{i\in(1,2\cdots ,n)|x_i=x_k\}$. Then in Proposition 1 the author claims $x\in (\mathbb{P^1})^n$ is semistable iff for all $k$, $\sum_{i\in I_k(x)}m_i\leq\frac{|m|}{2}$. The author has not given any proof and I cannot do it myself. Any proof or comment will be highly appreciated.

Answer 1

The details of this proof can be found in introductory texts on geometric invariant theory (GIT). I recommend Dolgachev's textbook Lectures on Invariant Theory, Chapter 11, where one can find a proof via the Hilbert-Mumford numerical criterion. The technical overhead to learning the Hilbert-Mumford numerical criterion is a bit high, but the benefit is that the same technique applies to many other GIT problems.

The only drawback of Dolgachev's treatment is that it steps through a chain of inequalities that, to a student, will probably appear unmotivated. I present a geometrically motivated sketch of the proof below. The full proof along these lines is given in Theorem 3.6 of a preprint of mine. (Many thanks to user afh for linking to it in the comments!)

Informally, one expects an object in projective space to be GIT unstable if that object has a high order of contact with a complete flag in projective space. In low-dimensional situations, instead of talking about flags, it may be enough to talk about order of contact with points and lines. For instance, a plane cubic curve is non-stable if it is singular, and having a singularity is the same as having a point of higher-than-expected multiplicity. In the case of configurations of points on $\mathbb{P}^1$, this philosophy says that if the configuration has a point of high multiplicity, the configuration should be unstable.

The way we compute this ``order of contact'' between a complete flag and our object, in practice, is with the Hilbert-Mumford criterion, and the closely related concept of the weight polytope (Chapter 9 in Dolgachev's textbook). In your setting, the case of an ${\text{SL}}_2$-action acting linearly on some variety $V$ embedded in projective space over $\mathbb{C}$ via some ample ${\text{SL}}_2$-linearized line bundle $\mathcal{L}$, the data of a complete flag is nothing more than a choice of 1-dimensional subspace of $\mathbb{C}^2$. To any complete flag $F$ in $\mathbb{C}^2$ and any point $v \in V$, we can associate a weight polytope $\Pi(v, F, \mathcal{L})$. The point $v \in V$ is semistable for $\mathcal{L}$ if and only if, for every flag $F$, the weight polytope $\Pi(v, F, \mathcal{L})$ contains the origin.

If ${\text{SL}}_2$ acts on two varieties $V$ and $W$, then the weight polytopes for the product action on $V \times W$, as embedded in projective space with the Segre embedding, are computed by $$\Pi((v,w),F, \mathcal{L}) = \Pi(v,F, \mathcal{L}) + \Pi(w,F, \mathcal{L}),$$ where $+$ denotes Minkowski sum. Also, the linear ${\text{SL}}_2$-action on $V$ induces ${\text{SL}}_2$-actions on the Veronese embeddings of $V$ in larger projective spaces, and this scales the weight polytope: for any $m \in \mathbb{N}$, we have $$\Pi(v,F, \mathcal{L}^{\otimes m}) = m \Pi(v, F, \mathcal{L}).$$

This is good news for your problem, because it means that you just need to understand the possible weight polytopes for the ${\text{SL}}_2$-action on $\mathbb{P}^1$ and sheaf $\mathcal{O}(1)$. The weight polytope in this case is either a point or a closed interval in $\mathbb{R}$, and it can be computed as follows. Given a flag in $\mathbb{C}^2$ associated to a one-dimensional subspace $H$ of $\mathbb{C}^2$, and a point $v \in \mathbb{P}^1,$ there are three possibilities for the weight polytope $\Pi = \Pi(v, H, \mathcal{O}(1))$. If $v \in H$, then $\Pi = \{1\}$; if $v \in H^\perp$, then $\Pi = \{-1\}$; otherwise $\Pi = [-1,1]$. Returning to the case of your problem, when we take Minkowski sums of such intervals with equal weights, the only way that the flag will detect instability is if more than $n/2$ points lie on $H$ or $H^\perp$. With possibly distinct weights, one may still compute the endpoints of the weight polytope to get the result.

I suggest carefully tracing through the definition of weight polytope in Dolgachev's book to understand the calculation above. The three possibilities for $\Pi$ correspond to the possible forms of the coordinates of $v$ when written in a basis with first vector taken from $H$.

I'll also mention that the problem of finding quotients of spaces of $n$ points on $\mathbb{P}^1$ was also addressed in the foundational text Geometric Invariant Theory (Chapter 3), as well as in the more approachable Introduction to the Theory of Moduli by Mumford and Suominen (Section 2).