Compact moduli of K3 surfaces

Abstract

Let F be a moduli space of lattice-polarized K3 surfaces and suppose that one has chosen a canonical effective ample divisor on a general fiber. We call this divisor "recognizable" if its flat limit on Kulikov surfaces is well defined. We prove that the normalization of the stable pair compactification F R for a recognizable divisor R is semitoroidal.We also prove that for polarized K3 surfaces (X, L) of degree 2d the sum R of rational curves in the linear system |L| is a recognizable divisor, giving a modular semitoroidal compactification of F 2d for all d.

Finite subgroups of automorphisms of K3 surfaces

“…First note that for y ∈ V we have τ y (L) ⊆ L if and only if y ∈ Λ. This shows part (1). Any isometry of L extends uniquely to an isometry of V and is-if the determinant is not 1-thus of the form στ α for some α ∈ V .…”

Compact moduli of K3 surfaces with a nonsymplectic automorphism

“…But it uses [Dol96, Thm. 3.1] which unfortunately is false, as was noted in [AE21]. For this reason, we decided to give an alternative construction.…”

“…Recognizable divisors. We recall the main new concept "recognizability" introduced in [AE21]. We slightly modify the definition as necessary for moduli spaces of K3 surfaces with ρ-markable automorphism: Definition 3.22.…”

“…4.18], is simply the first partial quotient D ρ /U K can . [AE21,Prop. 4.16] shows that D(J) embeds into a family of affine lines over J ⊥ /J ⊗ Z E where E is the universal elliptic curve over H (−H) and D(J) λ is its closure in a projective line bundle.…”

The flex divisor of a K3 surface

“…By the main theorem of [AE21], the normalization of F R 2d is semitoroidal whenever R satisfies a property called recognizability. Thus, the search for modular toroidal compactifications of F 2d is intimately related to finding canonical choices of polarizing divisor, and verifying that those choices are recognizable.…”