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Type II Degenerations of K3 Surfaces of Degree 4

James Matthew Jones

TL;DR

This work classifies Type II degenerations of K3 surfaces of degree $4$ via Tyurin degenerations, constructing explicit geometric models for each 1-dimensional boundary component of the Baily-Borel compactification $\overline{\mathcal{F}_4}$ and organizing them into $9$ lattice types. By analyzing the period map for Tyurin Kulikov surfaces and using nef/divisor models, the authors produce explicit degenerations with central fibers $V_0\cup V_1$, compute their stable models, and establish 18-dimensional families for each boundary component linked to the lattices $I^{\perp}/I$. They develop a comprehensive framework connecting explicit geometric degenerations (including non-normal quartics and weighted blowups) to the boundary components, and formulate Type II $\lambda$-families over tubular neighborhoods with surjective period maps. The results yield a complete dictionary between GIT degenerations, period-domain boundary data, and birational stable models for degree $4$ K3s, including nef-model decompositions and wall-chamber structures that organize the lifted polarizations. Overall, the paper provides an explicit, lattice-guided catalog of Tyurin degenerations, clarifying how period information distinguishes boundary components and how stable models arise from angular variations of polarizations.

Abstract

We study Type II degenerations of K3 surfaces of degree 4 where the central fiber consists of two rational components glued along an elliptic curve. Such degenerations are called Tyurin degenerations. We construct explicit Tyurin degenerations corresponding to each of the 1-dimensional boundary components of the Baily-Borel compactification of the moduli space of K3 surfaces of degree 4. For every such boundary component we also construct an 18-dimensional family of Tyurin degenerations of K3 surfaces of degree 4 and compute the stable models of these degenerations.

Type II Degenerations of K3 Surfaces of Degree 4

TL;DR

This work classifies Type II degenerations of K3 surfaces of degree via Tyurin degenerations, constructing explicit geometric models for each 1-dimensional boundary component of the Baily-Borel compactification and organizing them into lattice types. By analyzing the period map for Tyurin Kulikov surfaces and using nef/divisor models, the authors produce explicit degenerations with central fibers , compute their stable models, and establish 18-dimensional families for each boundary component linked to the lattices . They develop a comprehensive framework connecting explicit geometric degenerations (including non-normal quartics and weighted blowups) to the boundary components, and formulate Type II -families over tubular neighborhoods with surjective period maps. The results yield a complete dictionary between GIT degenerations, period-domain boundary data, and birational stable models for degree K3s, including nef-model decompositions and wall-chamber structures that organize the lifted polarizations. Overall, the paper provides an explicit, lattice-guided catalog of Tyurin degenerations, clarifying how period information distinguishes boundary components and how stable models arise from angular variations of polarizations.

Abstract

We study Type II degenerations of K3 surfaces of degree 4 where the central fiber consists of two rational components glued along an elliptic curve. Such degenerations are called Tyurin degenerations. We construct explicit Tyurin degenerations corresponding to each of the 1-dimensional boundary components of the Baily-Borel compactification of the moduli space of K3 surfaces of degree 4. For every such boundary component we also construct an 18-dimensional family of Tyurin degenerations of K3 surfaces of degree 4 and compute the stable models of these degenerations.

Paper Structure

This paper contains 21 sections, 56 theorems, 76 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Description of the Results
  4. Relation to Previous Work
  5. Data Availability
  6. Acknowledgements
  7. Background Material
  8. The Period Map for Kulikov Surfaces
  9. Type II $\lambda$-families
  10. Explicit Descriptions of the Families
  11. Two Quadrics Intersecting Transversally
  12. A Plane Intersecting a Cubic
  13. A Surface Non-Normal Along a Conic
  14. A Surface Non-Normal Along a Line
  15. A Surface Non-Normal Along a Twisted Cubic
  16. ...and 6 more sections

Key Result

Theorem 1

Let $X \rightarrow \Delta$ be a degeneration of K3 surfaces of degree 4 whose central fiber $X_0$ is a surface belonging to column 1 of Table tab:correspondence and let $\widetilde{X}$ be the minimal resolution of $X$. Then $\widetilde{X}$ is a projective Tyurin degeneration of K3 surfaces of degree

Figures (1)

  • Figure 1: The wall and chamber decompositions for $\operatorname{Lift}_{\geq0}(X_0)$ for each case of degeneration.

Theorems & Definitions (133)

  • Theorem 1
  • Definition 1.2
  • Theorem 2
  • Corollary 3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Strong Torelli
  • Definition 2.5
  • Definition 2.6
  • ...and 123 more