2023/09/19 by Pan Long, Long Pan, Fei Si +5
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AG
paper · pdf · doi:10.48550/arxiv.2309.10467
41 pages, the paper is re-organised and some errors fixed. Comments are very welcome!
openalex publication_date 2023/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01 · arxiv created 2026/08/03 · arxiv updated 2026/08/04
In this paper, we investigate the geometry of moduli space Pd of degree d del Pezzo pair, that is, a del Pezzo surface X of degree d with a curve C ∼ -2KX. More precisely, we study compactifications for Pd 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 Pd. For d=8 case, we propose the Hassett-Keel-Looijenga program \cF8(s)=\Proj(R(\cF8,Δ(s) ) as the section rings of certain \bQ-line bundle Δ8(s) on locally symmetric variety \cF8, which is birational to P8. Moreover, we give an arithmetic stratification on \cF8. After using the arithmetic computation of pullback Δ(s) on these arithmetic strata, we give the arithmetic predictions for the wall-crossing behavior of \cF8(s) when s∈ [0,1] varies. The relation of \cF8(s) with the K-moduli spaces of degree 8 del Pezzo pairs is also proposed.