K-moduli spaces of log Del Pezzo pairs

Abstract

We establish the full explicit wall-crossings for K-moduli space ${\bar{P}}^K\left(c\right)$ of degree 8 del Pezzo pairs $(X,cC)$, where generically $X\cong F_1$ and $C\sim -2K_X$. We also show that the K-moduli spaces ${\bar{P}}^K\left(c\right)$ coincide with the Hassett-Keel-Looijenga (HKL) models $F\left(s\right)$ of an 18-dimensional locally symmetric space associated with the lattice $E_8\oplus U^2\oplus E_7\oplus A_1$ under the transformation $s\left(c\right)=\frac{1-2c}{56c-4}$. This implies that the K-moduli spaces interpolate the GIT partial compactification and the Baily-Borel compactification for the moduli space of smooth Del Pezzo pairs. Some discussions concerning the relationship to KSBA moduli spaces are also provided.