Birational geometry of moduli space of del Pezzo pairs

Abstract

In this paper, we investigate the geometry of the moduli space $P_d$ of degree $d$ smooth del Pezzo pairs, which consists of a smooth del Pezzo surface $X$ of degree $d$ and a smooth curve $C\sim -2K_X$. More precisely, we study the compactifications of $P_d$ from both Hodge-theoretic and geometric invariant theoretical (GIT) perspectives. We obtain the class numbers of the Baily-Borel compactification $P_d^{\ast }$ for $P_d$, which is an important step toward establishing the Hassett-Keel-Looijenga program for $P_d$. If $d=8$, $P_d$ has two connected components. For the component parametrizing del Pezzo pairs $(Bl_p \PP^2, C)$, we propose the Hassett-Keel-Looijenga models $\cF(s)=\proj R(\cF,Δ(s) )$ via the section rings of certain $\bQ$-line bundles $\Delta (s)$ on the locally symmetric variety $\cF$. These models are expected to connect different birational models of the moduli space arising from K-moduli theory. By constructing an arithmetic stratification on $\cF$ and computing the pullback of $\Delta (s)$ on these strata, we give arithmetic predictions for the wall-crossing of $\cF(s)$ as $s\in [0,1]$ varies. This work parallels that of Laza-O'Grady \cite{LO19, LaO18}.

41 pages, the paper is re-organised and some errors fixed. Comments are very welcome!