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 P³ 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 P¹ P¹. As an application, we show that our K-moduli space appears as one model of the Hassett--Keel program for M₄. 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.