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Line configurations and K3 surfaces

Elias Sink

TL;DR

The paper studies realization spaces of $10_3$ line configurations, proving that rational realizations are dense in real realizations for all configurations and showing that four configurations admit Calabi–Yau-type compactifications by elliptic K3 surfaces with Picard number $20$ and specific discriminants. By constructing elliptic fibrations and analyzing their Mordell–Weil groups, the authors identify the K3 surfaces (notably showing $\widetilde{S}_{\mathrm V}$ equals the universal elliptic curve over $\Gamma_1(7)$) and establish a moduli interpretation via geometric invariant theory, with the democratic weight yielding a fine moduli K3 surface $\mathscr R_{\delta}(\mathscr L_N)$. These results connect combinatorial line configurations to modular elliptic surfaces and provide an explicit moduli-theoretic framework for the realizations. The work combines explicit surface geometry, lattice-theory invariants, and computational tools to elucidate when realization spaces attain K3-compactifications and how they can be interpreted as moduli spaces.

Abstract

We study the realization spaces of $10_3$ line configurations. Answering a question posed by Sturmfels in 1991, we use elliptic surface techniques to show that realizations over $\mathbb{Q}$ are dense in those over $\mathbb{R}$ for all $10_3$ configurations. We find that for exactly four of the ten configurations, the realization space admits a compactification by a K3 surface. We show that these have Picard number 20 and compute their discriminants. Finally, we use geometric invariant theory to give an elegant interpretation of these K3 surfaces as moduli spaces.

Line configurations and K3 surfaces

TL;DR

The paper studies realization spaces of line configurations, proving that rational realizations are dense in real realizations for all configurations and showing that four configurations admit Calabi–Yau-type compactifications by elliptic K3 surfaces with Picard number and specific discriminants. By constructing elliptic fibrations and analyzing their Mordell–Weil groups, the authors identify the K3 surfaces (notably showing equals the universal elliptic curve over ) and establish a moduli interpretation via geometric invariant theory, with the democratic weight yielding a fine moduli K3 surface . These results connect combinatorial line configurations to modular elliptic surfaces and provide an explicit moduli-theoretic framework for the realizations. The work combines explicit surface geometry, lattice-theory invariants, and computational tools to elucidate when realization spaces attain K3-compactifications and how they can be interpreted as moduli spaces.

Abstract

We study the realization spaces of line configurations. Answering a question posed by Sturmfels in 1991, we use elliptic surface techniques to show that realizations over are dense in those over for all configurations. We find that for exactly four of the ten configurations, the realization space admits a compactification by a K3 surface. We show that these have Picard number 20 and compute their discriminants. Finally, we use geometric invariant theory to give an elegant interpretation of these K3 surfaces as moduli spaces.
Paper Structure (5 sections, 17 theorems, 34 equations, 1 figure, 1 table)

This paper contains 5 sections, 17 theorems, 34 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Elliptic fibrations and density of rational realizations
  3. The K3 surfaces
  4. Moduli space interpretation
  5. Computations

Key Result

Theorem 1.1

The rational realizations $\mathscr R(\mathscr L)(\mathbb{Q})$ are Zariski-dense in $\mathscr R(\mathscr L)(\mathbb{R})$ for all $10_3$ configurations $\mathscr L$, and dense in the analytic topology for $\mathscr L\neq \mathscr{L}_{\mathrm{X}}$.

Figures (1)

  • Figure 1: The elliptic fibrations of $S$ and $\widetilde{S}$, drawn over $\mathbb{R}$ with coordinate $t=u/v$ on $\mathbb{P}^1$. The marked points are Du Val singularities of $S$ and are replaced by a chain of rational curves in $\widetilde{S}$. Smooth fibers, as well as the nodal fibers (not pictured), are unchanged by $\varphi$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • proof
  • ...and 29 more