Abstract

We describe a compactification by KSBA stable pairs of the five-dimensional moduli space of K3 surfaces with a purely non-symplectic automorphism of order four and$U(2)\oplus D_4^{\oplus 2}$lattice polarization. These K3 surfaces can be realized as the minimal resolution of the double cover of$\mathbb {P}^{1}\times \mathbb {P}^{1}$branched along a specific$(4,\,4)$curve. We show that, up to a finite group action, this stable pairs compactification is isomorphic to Kirwan's partial desingularization of the GIT quotient$(\mathbb {P}^{1})^{8}{/\!/}\mathrm {SL}_2$with the symmetric linearization.

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