We show that the K-moduli spaces of log Fano pairs $\left(\mathbb {P}^1\times \mathbb {P}^1, cC\right)$ , where C is a $(4,4)$ curve and their wall crossings coincide with the VGIT quotients of $(2,4)$ , complete intersection curves in $\mathbb {P}^3$ . 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 $\mathbb {P}^1\times \mathbb {P}^1$ and the Baily–Borel compactification of moduli of quartic hyperelliptic K3 surfaces.

我们证明了 log Fano 对的 K 模空间 $\left(\mathbb {P}^1\times \mathbb {P}^1, cC\right)$ ，其中C是 $(4,4)$ 曲线及其壁交叉点与 $(2,4)$ 的 VGIT 商重合，在 $\mathbb {P}^3$ 中完成相交曲线。这与 Laza 和 O'Grady 最近的结果一起，意味着这些 K 模空间在 $(4,4)$ 曲线的 GIT 模空间之间形成自然插值 $\mathbb {P}^1\times \ mathbb {P}^1$ 和四次超椭圆 K3 曲面模量的 Baily–Borel 紧化。