Automorphisms of a $(24_4, 32_3)$-configuration on the Schur quartic

Abstract: We answer the automorphism question raised by Naskręcki and Pokora for their $(24_4, 32_3)$-configuration on the Schur quartic. The stabilizer of either $24$-line half is isomorphic to $W(D_4)\rtimes C_3$, where $C_3$ acts by even triality; the full projective automorphism group has order $1152$. The $D_4$ model also gives an intrinsic coloring and shortens the incidence check.

• The paper establishes that while the full automorphism group of the Schur quartic is infinite, its projective automorphism group stabilizing the hyperplane class is finite with order 1152.
• It uses a root-theoretic labeling of the 24-line configuration to identify stabilizer subgroups, notably showing an isomorphism to W(D4)⋊C3 that encodes key combinatorial incidences.
• The study offers practical methods for streamlining symmetry verifications on K3 surfaces by reducing extensive pairwise checks through orbit analysis.

Automorphism Groups of a $(24_4, 32_3)$-Configuration on the Schur Quartic

Introduction

The work systematically addresses the automorphism structure of the $(24_4, 32_3)$ configuration on the Schur quartic surface, resolving specific open questions concerning group actions raised by Naskręcki and Pokora in their analysis of special line arrangements on quartic surfaces. The Schur quartic's deep connections to singular $K3$ surfaces, root systems, and exceptional finite groups set the foundational context for this study.

Geometric and Arithmetic Framework

The Schur quartic, embedded as the surface $$X = \{ x_0^4 - x_0x_1^3 - x_2^4 + x_2x_3^3 = 0 \} \subset \mathbb{P}^3$$, carries 48 lines of the second kind, decomposable into two natural disjoint 24-line sets, $D$ and $D^*$. Each set encodes a $(24_4, 32_3)$ configuration, where each line meets four others and is contained in exactly three distinguished triples. This geometric realization is algebraically modeled by divisor classes: $D \sim 6H$, ${_2} := D + D^* \sim 12H$ (with $H$ the hyperplane section).

Through the work of Degtyarev, the surface is identified with the singular $(24_4, 32_3)$0 surface $(24_4, 32_3)$1 with transcendental lattice isomorphic to $(24_4, 32_3)$2. The Néron-Severi group is $(24_4, 32_3)$3, and the Picard number is maximal, $(24_4, 32_3)$4. This guarantees the absence of nontrivial torsion in the Néron-Severi group, crucial for automorphism lifting arguments.

Main Results: Automorphism and Stabilizer Structure

The central outcomes can be summarized as follows:

• The full abstract automorphism group $(24_4, 32_3)$5 of the Schur quartic is infinite; however, the projective automorphism group stabilizing the hyperplane class $(24_4, 32_3)$6—that is, projective automorphisms $(24_4, 32_3)$7—forms a finite group $(24_4, 32_3)$8 of order $(24_4, 32_3)$9, specifically $K3$0. The group fits precisely case 77a in Brandhorst-Hashimoto's classification ([BH21]).
• The stabilizer of either 24-line half (e.g., $K3$1 or $K3$2) is the group $K3$3 of order 576, where $K3$4 acts via even triality on the Weyl group $K3$5. This subgroup is described algebraically as $K3$6 (with $K3$7 the binary tetrahedral group). The intrinsic labeling of lines by $K3$8 roots enables reconstruction of all incidence relations and inner products combinatorially.
• Group extension structure: The projective automorphism group $K3$9 fits into the exact sequence

$$X = \{ x_0^4 - x_0x_1^3 - x_2^4 + x_2x_3^3 = 0 \} \subset \mathbb{P}^3$$0

encoding the interchange of the two 24-line halves by the quotient $$X = \{ x_0^4 - x_0x_1^3 - x_2^4 + x_2x_3^3 = 0 \} \subset \mathbb{P}^3$$1.

Incidence Structure, Root Systems, and Shortcuts

The root-theoretic labeling provides a canonical identification of the 24 lines with the 24 roots of type $$X = \{ x_0^4 - x_0x_1^3 - x_2^4 + x_2x_3^3 = 0 \} \subset \mathbb{P}^3$$2. The 32 distinguished triples emerge as unordered root-triples summing to zero. The automorphism group $$X = \{ x_0^4 - x_0x_1^3 - x_2^4 + x_2x_3^3 = 0 \} \subset \mathbb{P}^3$$3 acts on this set, but only the even triality subgroup $$X = \{ x_0^4 - x_0x_1^3 - x_2^4 + x_2x_3^3 = 0 \} \subset \mathbb{P}^3$$4 preserves the coloring and hence corresponds to $$X = \{ x_0^4 - x_0x_1^3 - x_2^4 + x_2x_3^3 = 0 \} \subset \mathbb{P}^3$$5.

Orbit analysis under $$X = \{ x_0^4 - x_0x_1^3 - x_2^4 + x_2x_3^3 = 0 \} \subset \mathbb{P}^3$$6 reveals four classes on pairs of lines, with sizes determined by the $$X = \{ x_0^4 - x_0x_1^3 - x_2^4 + x_2x_3^3 = 0 \} \subset \mathbb{P}^3$$7 root inner product: $$X = \{ x_0^4 - x_0x_1^3 - x_2^4 + x_2x_3^3 = 0 \} \subset \mathbb{P}^3$$8 (12), $$X = \{ x_0^4 - x_0x_1^3 - x_2^4 + x_2x_3^3 = 0 \} \subset \mathbb{P}^3$$9 (72), $D$0 (96), $D$1 (96). This provides an immediate combinatorial shortcut, bypassing checks on all $D$2 line pairs and $D$3 triples in verifying the configuration.

The natural bipartition of the 48 lines into the two 24-line components corresponds precisely to the two connected components of the triple-incidence graph. This coloring is unique up to exchange and aligns with the intrinsic combinatorics of the configuration, as conjectured by Naskręcki and Pokora.

Theoretical and Practical Implications

This analysis rigorously determines the complete algebraic automorphism structure for the $D$4 configuration, settling questions of uniqueness, symmetry, and canonical coloring raised in recent work. By linking configuration automorphisms with explicit finite group actions on root systems, the paper strengthens the bridge between finite geometry, algebraic surfaces, and group theory.

Practically, these results allow for significant streamlining of combinatorial or geometric verification procedures in computational models of $D$5 surfaces, as well as for future algebraic and arithmetic investigations of symmetry properties in high Picard number $D$6's. The identification of the automorphism group as a subgroup of $D$7 and its explicit triality action may inspire novel connections with lattice theory, especially concerning moduli of quartic surfaces and their associated line bundles.

The approach foregrounds the utility of root/weight system labelings in the analysis of line configurations, promising adaptations in corresponding contexts (e.g., other quartics, del Pezzo surfaces, and lattice-polarized $D$8's). The direct verification methods suggested have relevance for algorithmic implementation in algebraic geometry software, reducing computational complexity.

Future Directions

Key avenues opened include the study of automorphism-induced dynamics on the Néron-Severi and transcendental lattices, further exploitation of root-theoretic symmetries for classifying other line or rational curve configurations on $D$9 surfaces, and the investigation of arithmetic consequences related to moduli and mod-$D^*$0 reductions. Analyzing how these automorphism structures interact with degenerations or deformations of the Schur quartic, or with broader mirror symmetry phenomena, constitutes a promising direction.

Conclusion

The paper provides a definitive algebraic and combinatorial characterization of the automorphism group structures associated with the $D^*$1-configuration on the Schur quartic. By identifying the stabilizers as concrete extensions of finite simple groups and embedding the geometry within the context of $D^*$2 root systems, it resolves longstanding questions and establishes techniques with broad applicability in algebraic geometry and the theory of $D^*$3 surfaces (2607.10090).