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Compactifications of moduli spaces of K3 surfaces with a higher-order nonsymplectic automorphism

Valery Alexeev, Anand Deopurkar, Changho Han

TL;DR

This work provides a comprehensive description of modular compactifications for moduli spaces of K3 surfaces equipped with a purely non-symplectic automorphism of order $N\in\{3,4\}$, focusing on fixed loci that include a genus-$g$ curve with $g\ge 2$. The authors unify Hodge-theoretic (BB and toroidal) and algebro-geometric (KSBA stable-pair) compactifications by establishing a KSBA semifan and relating it to semi-toroidal boundary data via ADE-root lattices, Eisenstein lattices, and Kulikov degenerations. For $N=4$ they show the KSBA moduli space coincides with a semi-toroidal compactification having a single BB cusp; for $N=3$ they classify four maximal families and compute their KSBA semifans explicitly, using the triple Tschirnhausen construction to produce and control a wide range of Kulikov degenerations. The results bridge lattice-theoretic boundary phenomena with geometric degenerations of triple covers, enabling a precise modular interpretation of the boundary components and their contractions, and they connect to the moduli of stable log quadrics and related Hurwitz-type spaces. This yields a detailed, lattice-driven picture of how KSBA stability mirrors toroidal compactifications in families of K3 surfaces with higher-order nonsymplectic automorphisms, with concrete, case-by-case boundary descriptions across several moduli spaces.

Abstract

We describe Baily-Borel, toroidal, and geometric -- using the KSBA stable pairs -- compactifications of some moduli spaces of K3 surfaces with a nonsymplectic automorphism of order $3$ and $4$ for which the fixed locus of the automorphism contains a curve of genus $\ge2$. For order $3$, we treat all the maximal-dimensional such families. We show that the toroidal and the KSBA compactifications in these cases admit simple descriptions in terms of certain $ADE$ root lattices.

Compactifications of moduli spaces of K3 surfaces with a higher-order nonsymplectic automorphism

TL;DR

This work provides a comprehensive description of modular compactifications for moduli spaces of K3 surfaces equipped with a purely non-symplectic automorphism of order , focusing on fixed loci that include a genus- curve with . The authors unify Hodge-theoretic (BB and toroidal) and algebro-geometric (KSBA stable-pair) compactifications by establishing a KSBA semifan and relating it to semi-toroidal boundary data via ADE-root lattices, Eisenstein lattices, and Kulikov degenerations. For they show the KSBA moduli space coincides with a semi-toroidal compactification having a single BB cusp; for they classify four maximal families and compute their KSBA semifans explicitly, using the triple Tschirnhausen construction to produce and control a wide range of Kulikov degenerations. The results bridge lattice-theoretic boundary phenomena with geometric degenerations of triple covers, enabling a precise modular interpretation of the boundary components and their contractions, and they connect to the moduli of stable log quadrics and related Hurwitz-type spaces. This yields a detailed, lattice-driven picture of how KSBA stability mirrors toroidal compactifications in families of K3 surfaces with higher-order nonsymplectic automorphisms, with concrete, case-by-case boundary descriptions across several moduli spaces.

Abstract

We describe Baily-Borel, toroidal, and geometric -- using the KSBA stable pairs -- compactifications of some moduli spaces of K3 surfaces with a nonsymplectic automorphism of order and for which the fixed locus of the automorphism contains a curve of genus . For order , we treat all the maximal-dimensional such families. We show that the toroidal and the KSBA compactifications in these cases admit simple descriptions in terms of certain root lattices.

Paper Structure

This paper contains 59 sections, 38 theorems, 100 equations, 24 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Moduli of K3 surfaces with a non-symplectic automorphism
  3. The period map
  4. The Baily--Borel compactification
  5. Semitoroidal compactifications
  6. KSBA stable pair compactifications
  7. Degenerations and Kulikov surfaces
  8. Kulikov models
  9. Nef, divisor, and stable models
  10. Periods of Type II Kulikov surfaces
  11. Moduli space of quadruple covers of P2
  12. Stable and Kulikov degenerations
  13. The KSBA semifan
  14. Eisenstein and 3-elementary lattices
  15. Generalities on Eisenstein lattices
  16. ...and 44 more sections

Key Result

theorem 1

For each of the four maximal families of K3 surfaces with a non-symplectic automorphism of degree 3 satisfying eq:g2, the compactification $\overline F_{\rho}^{\rm KSBA}$ is a semi-toroidal compactification of $\mathbb D_{\rho}/\Gamma_{\rho}$ given by the explicit semifan $\mathfrak{F}$ called the K

Figures (24)

  • Figure 1: We obtain a degeneration of ${\mathbb P}^2$ to $\mathop{\mathrm{Bl}}\nolimits_p{\mathbb P}(1,1,2) \cup {\mathbb P}(1,1,2)$ by blowing up succesively in two lines and blowing down a ${\mathbb P}^2$. The figure shows the central fibers in this process. The numbers next to the edges represent self-intersections. By taking a cyclic degree 4 cover, we obtain a type II Kulikov degeneration.
  • Figure 2: The pinched triple Tschirnhausen construction
  • Figure 3: In the case $m = 0$ and $C=(1,3)$, the surface $X$ (left) is the blow-up of a $\rm dP_3$$\overline X$ (right). The double curve is blue, the ramification curve is red, and the numbers are self-intersections.
  • Figure 4: In the case $m = 0$ and $C = (0,1) \sqcup (2,2)$, the surface $X$ (left) is the blow-up of a $\rm dP_1$$\overline X$ (right). The double curve is blue, the ramification curve is red, and the numbers indicate self-intersections.
  • Figure 5: In the case $m = 1$ and $C = (0,3)$, the surface $X$ (left) is the blow-up of ${\mathbb P}^2$ (right). The double curve is blue, the ramification curve is red, and the numbers indicate self-intersections.
  • ...and 19 more figures

Theorems & Definitions (86)

  • theorem 1
  • theorem 2
  • theorem 3
  • remark 4
  • theorem 5
  • definition 6
  • definition 7
  • definition 8
  • definition 9
  • lemma 10
  • ...and 76 more