On the moduli space of stable surfaces with p_g=1 realizing the minimal volume

2025/10/20 by Valery Alexeev, Alexeev, Valery, Wenfei Liu +3
Mathematics · #14D22 #14J28 #14J29 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2510.17678

openalex publication_date 2025/10/20 · openalex created_date 2025/10/22 · openalex updated_date 2026/07/28

Abstract

Let M₁ be the moduli space of the KSBA stable surfaces X of geometric genus p_g(X)=1 realizing the minimal possible volume K_X²=\frac1143. We show that its reduced part M_{1,\rm red} is a 10-dimensional projective variety isomorphic to the Baily--Borel compactification F_Λ^{\rm BB} of the moduli space of Λ-polarized K3 surfaces, where Λ=II_1,9≃ U⊕ E₈ is a unimodular lattice of signature (1,9). By a result of Brieskorn, F_Λ^{\rm BB} is a weighted projective space. We also verify the Viehweg hyperbolicity of the base of a Whitney equisingular family of stable surfaces in M₁. More generally, we prove that the same results hold for the moduli space M_c of KSBA stable pairs (X,B) with coefficients of B belonging to a set \mathcal C⊂ [0,1] such that \mathcal C∪\1\ attains a minimum, say c, and with p_g(X)=1, realizing the minimal possible volume (K_X+B)²=v(c). Indeed, we show that M_{c,\rm red} is independent of c and that for c≤\frac713 M_c is isomorphic to F_Λ^{\rm BB}.

Citations

• On lattice-polarized K3 surfaces
• Moduli of elliptic surfaces of Kodaira dimension one fibered over rational curves
• The Noether inequality for threefolds and three moduli spaces with minimal volumes
• Viehweg hyperbolicity for Whitney equisingular families with Gorenstein rational singularities
• Algebraic threefolds of general type with small volume
• Compact moduli of K3 surfaces
• Geometry of polarised varieties
• Moduli of polarized Calabi-Yau pairs
• Stable pair compactification of moduli of K3 surfaces of degree 2
• The minimal volume of log surfaces of general type with positive geometric genus
• Viehweg's hyperbolicity conjecture for families with maximal variation
• The ring of modular forms for the even unimodular lattice of signature (2,10)
• Fundamental theorems for semi log canonical pairs
• Mirror symmetry for lattice polarized K3 surfaces

Related

• The minimal and next minimal volumes of normal KSBA stable surfaces with p_g≥ 2
• KSBA compactification of the moduli space of K3 surfaces with purely non-symplectic automorphism of order four
• Moduli of weighted stable marked cubic surfaces
• Unimodal singularities and boundary divisors in the KSBA moduli of a class of Horikawa surfaces
• The KSBA compactification of the moduli space of D_1,6-polarized Enriques surfaces