Degenerations of elliptic quartics and K-moduli of Fano threefolds

We study the K-moduli space of Fano threefolds obtained by first blowing up $\mathbb{P}^3$ along an elliptic quartic curve $C$ and then blowing up a fiber $\ell_p$ of the exceptional divisor $E \to C$. We prove that this K-moduli space is isomorphic to a VGIT quotient parametrizing pairs $(C,p)$, with linearization induced by the CM line bundle. In particular, we classify all K-(semi/poly)stable members of this deformation family. The main new ingredients in the proof include the geometry of the Hilbert scheme of elliptic quartic curves, deformation theory of Fano--K3 pairs, and optimal bounds on the local volumes of threefold singularities.