The boundary of K-moduli of prime Fano threefolds of genus twelve

We study the K-moduli stack of prime Fano threefolds of genus twelve, known as $V_{22}$. We prove that its boundary, which parametrizes singular members, is purely divisorial and consists of four irreducible components corresponding to the four families of Prokhorov's one-nodal $V_{22}$. A key ingredient is a modular relation between Fano threefolds $X$ and their anticanonical K3 surfaces $S$. We prove that the forgetful morphism from the moduli of Fano--K3 pairs $(X,S)$ where $X$ is a K-semistable degeneration of $V_{22}$ to the moduli space of genus $12$ polarized K3 surfaces $(S,{-K_X}|_S)$ is an open immersion. In particular, the K-moduli of $V_{22}$ is governed by the moduli of their anticanonical K3 surfaces, providing a modular realization of Mukai's philosophy. Along the way, we develop a general deformation framework for Fano threefolds of large volume, which may be useful beyond the study of K-moduli.