K moduli of log del Pezzo pairs

Abstract: We establish the full explicit wall-crossing for K-moduli space $\overline{P}^K_c$ of degree $8$ del Pezzo pairs $(X,cC)$ where generically $X \cong \bbF_1$ and $C \sim -2K_X$. We also show K-moduli spaces $\overline{P}^K_c$ coincide with Hassett-Keel-Looijenga(HKL) models $\cF(s)$ of a $18$-dimensional locally symmetric spaces associated to the lattice $E_8\oplus U^2\oplus E_7\oplus A_1$.