Galois lines for a canonical curve of genus 4, III: non-cyclic Galois lines

Abstract: Let $C \subset \mathbb{P}<sup>3$ be a canonical curve of genus $4$ over an algebraically closed field $k$ of characteristic zero. For a line $l \subset \mathbb{P}<sup>3$, we consider the projection $π_l: C \to \mathbb{P}<sup>1$ from $l$ and the induced extension of function fields $$π_l^*: k(\mathbb{P}^{1)\hookrightarrow} k(C)$$. A line $l$ is called an \emph{$S_3$-line} (resp. a \emph{$K_4$-line}) if the extension $k(C)/π_l<sup>*(k(\mathbb{P}<sup>1))$ is Galois and its Galois group is isomorphic to the symmetric group $S_3$ on three letters (resp. the Klein four-group $K_4$). We prove that the number of $S_3$-lines (resp.\ $K_4$-lines) is at most $10$ (resp.\ $15$).

• The paper establishes sharp upper bounds by proving that there are at most 10 S₃-lines and 15 K₄-lines for canonical genus 4 curves.
• It employs group-theoretic techniques and explicit computational examples to classify Galois lines defined via canonical embeddings.
• The findings enhance understanding of automorphism groups and provide new methods for analyzing monodromy structures in algebraic curves.

Galois Lines for Canonical Curves of Genus 4: Non-Cyclic Galois Lines

Introduction and Context

This paper investigates the configuration and cardinality of non-cyclic Galois lines—specifically those with Galois groups $S_3$ and $K_4$—associated with canonical genus 4 curves over algebraically closed fields of characteristic zero. The context is the canonical embedding $C \subset \mathbb{P}^3$ of such curves, where the geometry of divisors and automorphism groups is rich and highly constrained.

A Galois line $l \subset \mathbb{P}^3$ for a curve $C$ is defined by the property that the projection $\pi_l: C \to \mathbb{P}^1$ induces a Galois extension of function fields $k(C)/\pi_l^*(k(\mathbb{P}^1))$. The nature of possible Galois groups arises from the geometry of $C$ (a $(2,3)$-complete intersection of degree 6), its automorphism group, and classical results on covers of curves.

Prior results have classified and bounded the number of cyclic Galois lines; the novel contribution of this work is a maximal upper bound for Galois lines with non-cyclic group, specifically $S_3$ (symmetric group on three elements) and $K_4$0 (Klein four-group).

Key Results and Theorems

The paper establishes the following central result:

• The number of $K_4$1-lines is at most 10.
• The number of $K_4$2-lines is at most 15.

Both bounds are shown to be sharp by explicit computational examples using a genus 4 curve with full automorphism group $K_4$3.

Structural and Technical Details

Canonical Curves of Genus 4

These curves are non-hyperelliptic and can be realized as $K_4$4-complete intersections in $K_4$5:

• If the bounding quadric $K_4$6 is singular ($K_4$7), the curve admits a unique trigonal morphism.
• If $K_4$8 is smooth ($K_4$9), the curve admits exactly two distinct trigonal morphisms.

The automorphism group acts naturally on $C \subset \mathbb{P}^3$0 and is realized via projective transformations.

Definition and Characterization of Galois Lines

For each line $C \subset \mathbb{P}^3$1, the projection $C \subset \mathbb{P}^3$2 and its associated field extension allows one to define Galois lines and determine possible Galois groups. Explicit matrix representations (via projective linear group elements) for the group actions are given, determined up to conjugacy.

• $C \subset \mathbb{P}^3$3-lines are associated with automorphism groups generated by diagonal and permutation matrices (as in Equation (2.3) of the paper).
• $C \subset \mathbb{P}^3$4-lines correspond to automorphism groups generated by two involutions with specific diagonal forms.

Distinct Galois lines correspond to automorphism subgroups that are not conjugate, and much of the analysis involves counting the number of conjugacy classes of such subgroups in $C \subset \mathbb{P}^3$5.

Bounding the Numbers: Group Theoretic Techniques

The paper uses several group-theoretic facts:

• Every finite subgroup of $C \subset \mathbb{P}^3$6 is cyclic, dihedral, or isomorphic to $C \subset \mathbb{P}^3$7, $C \subset \mathbb{P}^3$8, or $C \subset \mathbb{P}^3$9.
• For each possible finite subgroup, explicit counts are made of $l \subset \mathbb{P}^3$0 and $l \subset \mathbb{P}^3$1 subgroups, leading to the global bounds on Galois lines by reduction to combinatorial group theory.

The crucial argument is that for each Galois line, the Galois group $l \subset \mathbb{P}^3$2 injects into $l \subset \mathbb{P}^3$3 as a concrete permutation group with constraints on its fixed loci, which limits possibilities for overlap among Galois lines.

Explicit Example: The Maximal Automorphism Case

An explicit curve with $l \subset \mathbb{P}^3$4 is constructed. Using computer algebra (GAP), all the Galois lines with groups isomorphic to $l \subset \mathbb{P}^3$5 and $l \subset \mathbb{P}^3$6 are computed.

For this curve, the bounds are achieved ($l \subset \mathbb{P}^3$7 $l \subset \mathbb{P}^3$8-lines and $l \subset \mathbb{P}^3$9 $C$0-lines). Explicit equations for all these lines are given and the corresponding group actions realized as explicit $C$1 matrices.

Consequences and Implications

These results complete the classification of possible Galois lines (cyclic and non-cyclic) for canonical genus 4 curves regarding their possible Galois groups and the maximum number of lines with each group type.

The strong upper bounds further constrain the possible monodromy structures of space projections and offer finer insight into the interplay between the automorphism group and geometric function theory of these curves.

Notably, the construction and enumeration techniques provide a computational method for exploring related questions in higher genus or other moduli loci, leveraging the fact that in genus 4, the automorphism group structure is extremely rigid thanks to deep results of Kuribayashi & Kuribayashi.

Practically, the knowledge of all possible non-cyclic Galois lines is essential in the context of algebraic geometry, inverse Galois problems for function fields, and the explicit construction of covers with prescribed monodromy.

Theoretically, these results further demonstrate the subtle interaction between projective geometry, function field theory, and finite group actions, reinforcing the paradigm that combinatorial group theory can yield strong geometric conclusions for moduli spaces.

Outlook for Future Research

Future directions may involve analogous investigations for higher genus canonical curves, non-canonical embeddings, and related questions about the distribution and nature of Galois subcovers. The developed computational criteria signal potential for automated classification of Galois covers in computational algebraic geometry.

Another intriguing line of inquiry involves understanding how these geometrically defined maximal configurations relate to loci in moduli space with special automorphism group, and if the bounds can be improved or analogously characterized in positive characteristic or in the context of non-algebraically closed fields.

Conclusion

This paper gives a complete answer to the enumerative problem of non-cyclic Galois lines for canonical genus 4 curves, showing that at most ten $C$2-lines and fifteen $C$3-lines can exist, and that these bounds are realized for the unique maximal automorphism case. The work combines explicit geometric analysis, finite group theory, and computer-aided computation to achieve a definitive structural result on the geometry of space curves of low genus.