Birational geometry of moduli space of del Pezzo pairs

In this paper, we investigate the geometry of moduli space $P_d$ of degree $d$ del Pezzo pair, that is, a del Pezzo surface $X$ of degree $d$ with a curve $C \sim -2K_X$. More precisely, we study compactifications for $P_d$ from both Hodge's theoretical and geometric invariant theoretical (GIT) perspective. We compute the Picard numbers of these compact moduli spaces which is an important step to set up the Hassett-Keel-Looijenga models for $P_d$. For $d=8$ case, we propose the Hassett-Keel-Looijenga program $\cF_8(s)=\Proj(R(\cF_8,\Delta(s) )$ as the section rings of certain $\bQ$-line bundle $\Delta_8(s)$ on locally symmetric variety $\cF_8$, which is birational to $P_8$. Moreover, we give an arithmetic stratification on $\cF_8$. After using the arithmetic computation of pullback $\Delta(s)$ on these arithmetic strata, we give the arithmetic predictions for the wall-crossing behavior of $\cF_8(s)$ when $s\in [0,1]$ varies. The relation of $\cF_8(s)$ with the K-moduli spaces of degree $8$ del Pezzo pairs is also proposed.