K-Moduli of Curves on a Quadric Surface and K3 Surfaces

We show that the K-moduli spaces of log Fano pairs (P¹× P¹, cC) , where C is a $(4,4)$ curve and their wall crossings coincide with the VGIT quotients of $(2,4)$ , complete intersection curves in P³ . This, together with recent results by Laza and O’Grady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of $(4,4)$ curves on P¹× P¹ and the Baily–Borel compactification of moduli of quartic hyperelliptic K3 surfaces.