Birational geometry of moduli space of del Pezzo pairs

TL;DR

This paper explores the geometry and compactifications of the moduli space of degree d del Pezzo pairs, proposing models and stratifications, especially for the degree 8 case, with implications for wall-crossing and K-moduli spaces.

Contribution

It introduces new compactification models for the moduli space of del Pezzo pairs and analyzes their geometric and arithmetic properties, particularly for degree 8.

Findings

Computed Picard numbers of compact moduli spaces.

Proposed Hassett-Keel-Looijenga models for degree 8 case.

Predicted wall-crossing behavior of the models.

Abstract

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.