K3 surfaces with a symplectic action of (\mathbb Z/2\mathbb Z)²

2024/08/01 by Benedetta Piroddi, Piroddi, Benedetta
Mathematics · #14J27 #14J28 #14J50 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2408.00643

openalex publication_date 2024/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the symplectic action of the group (Z/2Z)² on a K3 surface X: we describe its action on H²(X,Z) and the maps induced in cohomology by the rational quotient maps; we give a lattice-theoretic characterization of the resolution of singularities of the quotient X/i, where i is any of the involutions in (Z/2Z)². Assuming X is projective, we describe the correspondence between irreducible components of its moduli space, and those of the resolution of singularities of its quotients: this being the first description of this correspondence for a non-cyclic action, we see new phenomena, of which we provide explicit examples assuming X has a polarization of degree 4.

Related

• K3 surfaces with ℤ₂² symplectic action
• K3 surfaces with a symplectic automorphism of order 4
• Kummer surfaces and K3 surfaces with (Z/2Z)⁴ symplectic action
• Symplectic automorphisms and the Picard group of a K3 surface
• K3 surfaces with action of the group M₂₀