K-moduli of Fano threefolds and genus four curves

In this article, we study the K-moduli space of Fano threefolds obtained by blowing up $\mathbb{P}^3$ along $(2,3)$-complete intersection curves. This K-moduli space is a two-step birational modification of the GIT moduli space of $(3,3)$-curves on $\mathbb{P}^1 \times \mathbb{P}^1$. As an application, we show that our K-moduli space appears as one model of the Hassett--Keel program for $\overline{M}_4$. In particular, we classify all K-(semi/poly)stable members in this deformation family of Fano varieties. We follow the moduli continuity method with moduli of lattice-polarized K3 surfaces, general elephants and Sarkisov links as new ingredients.