Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Normal Forms and Tyurin Degenerations of K3 Surfaces Polarised by a Rank 18 Lattice

Charles F. Doran, Joseph Prebble, Alan Thompson

TL;DR

The article classifies Tyurin degenerations of $M$-polarised K3 surfaces, showing that the central fibre geometry is not tightly constrained by lattice data: for the rank-18 lattice $M = H \oplus E_8 \oplus E_8$, there are 11 analytic central-fibre types, all of which can be realised as projective Tyurin degenerations connected by flopping curves. Using both linear-system and toric constructions, the authors produce explicit projective normal forms and degenerations, achieving a complete realisation table that includes del Pezzo components with all possible degrees $-9 \le K_{Y_i}^2 \le 9$ and various elliptic-glued configurations. The work also illuminates mirror-symmetry considerations via the DHT conjecture, concluding that lattice-polarisation data alone do not determine the mirror base-splitting or the geometry of the degenerations. Overall, the paper provides explicit, constructive models for all admissible degenerations and introduces new normal forms for $M$-polarised K3 surfaces.

Abstract

We study projective Type II degenerations of K3 surfaces polarised by a certain rank 18 lattice, where the central fibre consists of a pair of rational surfaces glued along a smooth elliptic curve. Given such a degeneration, one may construct other degenerations of the same kind by flopping curves on the central fibre, but the degenerations obtained from this process are not usually projective. We construct a series of examples which are all projective and which are all related by flopping single curves from one component of the central fibre to the other. Moreover, we show that this list is complete, in the sense that no other flops are possible. The components of the central fibres obtained include weak del Pezzo surfaces of all possible degrees. This shows that projectivity need not impose any meaningful constraints on the surfaces that can arise as components in Type II degenerations.

Normal Forms and Tyurin Degenerations of K3 Surfaces Polarised by a Rank 18 Lattice

TL;DR

The article classifies Tyurin degenerations of -polarised K3 surfaces, showing that the central fibre geometry is not tightly constrained by lattice data: for the rank-18 lattice , there are 11 analytic central-fibre types, all of which can be realised as projective Tyurin degenerations connected by flopping curves. Using both linear-system and toric constructions, the authors produce explicit projective normal forms and degenerations, achieving a complete realisation table that includes del Pezzo components with all possible degrees and various elliptic-glued configurations. The work also illuminates mirror-symmetry considerations via the DHT conjecture, concluding that lattice-polarisation data alone do not determine the mirror base-splitting or the geometry of the degenerations. Overall, the paper provides explicit, constructive models for all admissible degenerations and introduces new normal forms for -polarised K3 surfaces.

Abstract

We study projective Type II degenerations of K3 surfaces polarised by a certain rank 18 lattice, where the central fibre consists of a pair of rational surfaces glued along a smooth elliptic curve. Given such a degeneration, one may construct other degenerations of the same kind by flopping curves on the central fibre, but the degenerations obtained from this process are not usually projective. We construct a series of examples which are all projective and which are all related by flopping single curves from one component of the central fibre to the other. Moreover, we show that this list is complete, in the sense that no other flops are possible. The components of the central fibres obtained include weak del Pezzo surfaces of all possible degrees. This shows that projectivity need not impose any meaningful constraints on the surfaces that can arise as components in Type II degenerations.
Paper Structure (19 sections, 4 theorems, 85 equations, 5 figures, 1 table)

This paper contains 19 sections, 4 theorems, 85 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background on $M$-polarised K3 surfaces
  3. Lattice theory
  4. Geometry of $M$-polarised K3 surfaces
  5. Moduli of $M$-polarised K3 surfaces
  6. Tyurin degenerations
  7. Background
  8. Analytic Tyurin degenerations
  9. Linear system approach
  10. Toric approach
  11. Constructing Tyurin degenerations
  12. Constructions
  13. Central fibres $(V_8,V_{10})$ and $(V_9,V_9)$
  14. Central fibre $(\overline{V}_1,V_{17})$
  15. Central fibres $(V_7,V_{11})$, $(V_5,V_{13})$, and $(V_3,V_{15})$
  16. ...and 4 more sections

Key Result

Lemma 2.3

Let $X$ be a generic $M$-polarised K3 surface. Then the group of automorphisms which fix the $M$-polarisation has order $2$ and is generated by the fibrewise elliptic involution with respect to the standard fibration and its unique section $E_9$.

Figures (5)

  • Figure 2.1: Dual graph of the configuration of simple roots in $M$.
  • Figure 2.2: Dual graph of the $(-2)$-curves on $X$ along with the curves $S$ and $T$.
  • Figure 3.1: Dual graph of curves on $V_d$ for $d \geq 3$.
  • Figure 3.2: Limits of curves $E_i$ on $(V_6,V_{12})$. Curves on $V_6$ are to the left of the dashed edge and curves on $V_{12}$ are to the right. The dashed edge denotes the point $p \in D$. The curves $E_0,\ldots,E_5$ degenerate to $F_0,\ldots,F_5$, the curves $E_{7}\ldots,E_{18}$ degenerate to $F_{11}',\ldots,F_{0}'$, and the curve $E_6$ degenerates to $(F_6+F_{12}')$.
  • Figure 3.3: Dual graph of curves on the quotient by the fibrewise elliptic involution.

Theorems & Definitions (22)

  • Definition 2.1
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Definition 3.4
  • Lemma 3.5
  • proof
  • ...and 12 more