Compact moduli of K3 surfaces with a nonsymplectic automorphism

Abstract

We construct a modular compactification via stable slc pairs for the moduli spaces of K3 surfaces with a nonsymplectic automorphism under the assumption that the fixed locus of the automorphism contains a component of genus g ≥ 2, and prove that it is semitoroidal. Contents 1. Introduction 1 2. Moduli of K3s with a nonsymplectic automorphism 3 3. Stable pair compactifications 6 4. Moduli of quotient surfaces 13 References 14

Finite subgroups of automorphisms of K3 surfaces

“…Hence L = (στ α )(L) = σ(α)σ(L), that is, L and σ(L) are equivalent. This shows part (2). Now assume that L = ασ(L).…”

“…Thus e i L is orthogonal to e j L for i = j. Equation (1) yields the corresponding chain of finite index inclusions (2) f Γ L ⊆ L ⊆ ΓL.…”

“…We end this section by describing the computation of G L in case (2). Therefore let L be be an indefinite binary lattice over Z and…”

Unimodal singularities and boundary divisors in the KSBA moduli of a class of Horikawa surfaces

“…The interplay between geometric compactifications and Hodge theory is one of the driving forces in moduli theory. Wellstudied cases include abelian varieties [2], K3 surfaces [4][5][6][7][8], algebraic curves [13], cubic surfaces [28], and cubic fourfolds [47]. Recently, there has been a great focus on generalizing this interplay in the case of surfaces of general type-the so-called nonclassical cases [32,40].…”

“…𝐸12 𝑧 3 + 𝑦7 + 𝑎𝑦5 𝑧 𝑍 11 𝑦𝑧 3 + 𝑦5 + 𝑎𝑦4 𝑧 𝑊 12 𝑧 4 + 𝑦 5 + 𝑎𝑦 3 𝑧 2 𝐸 13 𝑧 3 + 𝑦 5 𝑧 + 𝑎𝑦 8 𝑍 12 𝑦𝑧 3 + 𝑦 4 𝑧 + 𝑎𝑦 3 𝑧 2 𝑊 13 𝑧 4 + 𝑦 4 𝑧 + 𝑎𝑦 6 𝐸 14 𝑧 3 + 𝑦 8 + 𝑎𝑦 6 𝑧 𝑍 13 𝑦𝑧 3 + 𝑦 6 + 𝑎𝑦 5 𝑧 Definition 2.4. Let 𝑋 be a variety and ∑ 𝑖 𝑏 𝑖 𝐵 𝑖 a ℚ-divisor on 𝑋 with 0 < 𝑏 𝑖 ≤ 1 and 𝐵 𝑖 prime divisors.…”