Optimal Degenerations of K-unstable Fano threefolds

We explicitly determine the optimal degenerations of Fano threefolds $X$ in family No 2.23 of Mori-Mukai’s list as predicted by the Hamilton-Tian conjecture. More precisely, we find a special degeneration $(\mathcal{X},{\xi }_0)$ of $X$ such that $({\mathcal{X}}_0,{\xi }_0)$ is weighted K-polystable, which is equivalent to $({\mathcal{X}}_0,{\xi }_0)$ admitting a Kähler-Ricci soliton (KRS) by \cite{HL23} and \cite{BLXZ23}. Furthermore, we study the moduli spaces of $({\mathcal{X}}_0,{\xi }_0)$. The $\mathbf{H}$-invariant of $X$ divides the natural parameter space into two strata, which leads to different moduli spaces of KRS Fano varieties. We show that one of them is isomorphic to the GIT-moduli space of biconic curves $C\subseteq {\mathbb{P}}^1\times {\mathbb{P}}^1$, and the other one is a single point.