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1. K-stability and KSBA stability


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      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?


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          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?


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              Problem 1.3.

              [Smyth] What is twisted $K$-stability/KSBA stability for the universal family $\pi:\mathcal{C}=\overline{\mathcal{M}}_{g,1}\to\overline{\mathcal{M}}_g$, $\mathcal{L}=\omega_\pi$, $D=\Delta_{1,1}$?


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                  Problem 1.4.

                  [Morrison] What are semistable limits of non-reduced quintic surfaces? For instance, $\lim_{t\to 0} (tF_5+F_3\cdot F_1^2)$?


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                      Problem 1.5.

                      [Liu] When is the set of K-semistable (singular) Fano varieties of dimension $n$ and volume $\geq c (>0)$ bounded? This is known for surfaces.
                        1. 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.