1. Kstability and KSBA stability

Problem 1.1.
[Fedorchuk, Laza] Is there a modification of $K$stability/KSBA stability such that curves with singularities “worse” than nodes (e.g. cusps, tacnodes) become stable? 
Problem 1.2.
[Coskun, Dervan] (Coskun) If $(X,L)$ is $K$stable, what can one say about the $\epsilon$ for which the perturbation $(X, L\epsilon D)$ remains $K$stable? (Dervan) If $X$ is a smooth projective variety with $K_X$ ample, for which $L\in\mathrm{Ample}(X)$ is $(X,L)$ $K$stable? What is the structure of this locus in $\mathrm{Ample}(X)$? Is the answer purely numerical? 
Problem 1.3.
[Smyth] What is twisted $K$stability/KSBA stability for the universal family $\pi:\mathcal{C}=\overline{\mathcal{M}}_{g,1}\to\overline{\mathcal{M}}_g$, $\mathcal{L}=\omega_\pi$, $D=\Delta_{1,1}$? 
Problem 1.4.
[Morrison] What are semistable limits of nonreduced quintic surfaces? For instance, $\lim_{t\to 0} (tF_5+F_3\cdot F_1^2)$? 
Problem 1.5.
[Liu] When is the set of Ksemistable (singular) Fano varieties of dimension $n$ and volume $\geq c (>0)$ bounded? This is known for surfaces.
Remark. [org.aimpl.user:yuchenl@math.princeton.edu] This problem was recently confirmed in any dimension by Chen Jiang in [arXiv:1705.02740].

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