3. GIT stability

Problem 3.1.
[Wang, Laza] (Question by Wang) Does the moduli space of Kstable del Pezzo surfaces have an asymptotic GIT description?
 (Conjecture by LazaOdaka) Asymptotic GIT for cubic surfaces coincides with ordinary GIT.
 (Question by LazaOdaka) Is $xyz+t^3=0$ asymptotically GIT stable?
 (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 $\overline{\mathcal{M}}_g(\mathbb{P}^1,d)\mathbin{/\mkern6mu/}PGL(2 ,\mathbb{C})$ 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 Kstability instead of Hilbert stability. Commented by Farkas: Define Kstability for syzygies.) 
Problem 3.5.
[Paul] Let $X$ be a smooth subvariety of $\mathbb{P}^n$ of dimension $d$. Let $\lambda$ be a oneparameter 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 $X\times\mathbb{P}^{d1}$ via the Segre embedding, $DF$ is the DonaldsonFutaki invariant, and $w_{\lambda}$ is the HilbertMumford weight. The above inequality is known for $d=n1$, 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.