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 ${\text{cF}}_8(s)=\text{Proj}(R({\text{cF}}_8,\mathrm{\Delta }(s))$ as the section rings of certain $\text{bQ}$-line bundle ${\mathrm{\Delta }}_8(s)$ on locally symmetric variety ${\text{cF}}_8$, which is birational to $P_8$. Moreover, we give an arithmetic stratification on ${\text{cF}}_8$. After using the arithmetic computation of pullback $\mathrm{\Delta }(s)$ on these arithmetic strata, we give the arithmetic predictions for the wall-crossing behavior of ${\text{cF}}_8(s)$ when $s\in [0,1]$ varies. The relation of ${\text{cF}}_8(s)$ with the K-moduli spaces of degree $8$ del Pezzo pairs is also proposed.