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) circle plus D-4(circle plus 2) lattice polarization. These K3 surfaces can be realized as the minimal resolution of the double cover of P-1 x 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 (P-1)(8)//SL2 with the symmetric linearization.
Moon, H.b., Schaffler, L. (2021). KSBA COMPACTIFICATION OF THE MODULI SPACE OF K3 SURFACES WITH A PURELY NON-SYMPLECTIC AUTOMORPHISM OF ORDER FOUR. PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY, 64(1), 99-127 [10.1017/S001309152100002X].