Autoequivalences and stability conditions on a degenerate K3 surface

We study autoequivalences and stability conditions on the derived category of coherent sheaves on a singular surface $X$ which arises as an open subvariety of a type III Kulikov degeneration of K3 surfaces. The surface $X$ consists of four irreducible components, one of which is $\mathbb{P}^2$, and the others are non-compact rational surfaces. Using a comparison with the total space of the degeneration, we show that the connected component $\mathrm{Stab}^\dagger(D^b_{\mathbb{P}^2}(X))$ of the space of stability conditions on the supported derived category $D^b_{\mathbb{P}^2}(X)$ containing geometric stability conditions is simply connected, and describe its wall-and-chamber structure via half-spherical twists. As consequences, we determine the subgroup of the autoequivalence group $\mathrm{Aut}(D^b(X))$ that preserves this component; it is isomorphic to $\mathbb{Z} \times \Gamma_1(3) \times \mathrm{Aut}(X)$, where $\Gamma_1(3) \subset \mathrm{SL}(2,\mathbb{Z})$ is the congruence subgroup of level~3.