
3. GIT stability

1. Problem 3.1.

[Wang, Laza]
1. (Question by Wang) Does the moduli space of K-stable del Pezzo surfaces have an asymptotic GIT description?
2. (Conjecture by Laza-Odaka) Asymptotic GIT for cubic surfaces coincides with ordinary GIT.
3. (Question by Laza-Odaka) Is $xyz+t^3=0$ asymptotically GIT stable?
4. (Question by Laza) Is there any example of asymptotic GIT for surfaces?
• Problem 3.2.

[Morrison] Can one exploit the convexity properties of the estimates in [MR3427574] for asymptotic stability for nodal curves to get more effective and sharper stability estimates, e.g. $5$-canonical curves?
• Problem 3.3.

[Kemeny] Describe the quotients $\overline{\mathcal{M}}_g(\mathbb{P}^1,d)\mathbin{/\mkern-6mu/}PGL(2 ,\mathbb{C})$ for suitable linearizations, characterizing the stable points. Do any of these quotients give versions of the space of admissible covers in which the base $\mathbb{P}^1$ does not break? Are any modular?
• Problem 3.4.

[Farkas, Dervan] (Farkas) Investigate Hilbert stability of syzygy point of generic canonical curves of genus $g$. More precisely, is the point $C\mapsto K_{p,1}(C,K_C) \in \mathrm{Grass}/SL_g$ stable for generic $[C]\in \overline{\mathcal{M}}_g$? The answer is yes if $p=1$ or $2$, so how about higher $p$? (Question by Dervan: Same question for K-stability instead of Hilbert stability. Commented by Farkas: Define K-stability for syzygies.)
• Problem 3.5.

[Paul] Let $X$ be a smooth subvariety of $\mathbb{P}^n$ of dimension $d$. Let $\lambda$ be a one-parameter subgroup of $SL(n+1,\mathbb{C})$. Does the following inequality $DF(\lambda)\leq w_{\lambda}(\Delta)-w_{\lambda}(R)$ hold? Here $R=\mathrm{Chow}(X)$, $\Delta$ is the dual variety of $X\times\mathbb{P}^{d-1}$ via the Segre embedding, $DF$ is the Donaldson-Futaki invariant, and $w_{\lambda}$ is the Hilbert-Mumford weight. The above inequality is known for $d=n-1$, but not known in general, even for $d=1$. Is it possible to find $X$ such that $DF(\lambda)\leq 0$ for any one parameter subgroup $\lambda$, but there exists a $\lambda$ such that $w_{\lambda}(\Delta) -w_{\lambda}(R)>0$? Such an $X$ would be $K$-stable but have no metric of constant scalar curvature. Is there a characterization of the $X$ for which the above inequality becomes an equality? A guess is that equality holds if and only if the limit cycle is reduced.

Cite this as: AimPL: Stability and moduli spaces, available at http://aimpl.org/stabmoduli.