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Smooth plane curves with a unique outer Galois point and their automorphism groups

Eslam Badr, Takeshi Harui

Abstract

We consider smooth plane curves $\mathcal{X}$ of degree $d\geq4$, defined over an algebraically closed field of characteristic $0$, that possess a unique outer Galois point. This geometric condition forces the curve to be a cyclic covering of the projective line, and ensures that its automorphism group fits into a specific theoretical framework. For each possible non-cyclic reduced automorphism group $\operatorname{Aut}_{\operatorname{red}}(\mathcal{X})$, we fully characterize the defining equation of $\mathcal{X}$ and the precise structure of its full automorphism group $\operatorname{Aut}(\mathcal{X})$. This comprehensive analysis not only identifies the exact form of the equation for each automorphism type but also establishes the detailed criteria under which these scenarios can occur, thereby offering a complete classification of defining equations for smooth plane curves with a unique outer Galois point and a non-cyclic reduced automorphism group.

Smooth plane curves with a unique outer Galois point and their automorphism groups

Abstract

We consider smooth plane curves of degree , defined over an algebraically closed field of characteristic , that possess a unique outer Galois point. This geometric condition forces the curve to be a cyclic covering of the projective line, and ensures that its automorphism group fits into a specific theoretical framework. For each possible non-cyclic reduced automorphism group , we fully characterize the defining equation of and the precise structure of its full automorphism group . This comprehensive analysis not only identifies the exact form of the equation for each automorphism type but also establishes the detailed criteria under which these scenarios can occur, thereby offering a complete classification of defining equations for smooth plane curves with a unique outer Galois point and a non-cyclic reduced automorphism group.

Paper Structure

This paper contains 21 sections, 33 theorems, 83 equations, 4 tables.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. The dihedral case $\operatorname{D}_m$
  4. Necessary conditions
  5. Sufficiency and group structure analysis
  6. Case $\delta=0$
  7. Case $\delta=1$
  8. Complete proof of Theorem \ref{['mainresultIII']}
  9. The symmetric group case $\operatorname{S}_4$
  10. Necessary conditions
  11. Sufficiency and group structure analysis
  12. Complete proof of Theorem \ref{['mainresultI']}
  13. The alternating group case $\operatorname{A}_5$
  14. Necessary conditions
  15. Sufficiency and group structure analysis
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\mathcal{X}:Z^d+L_{d,Z}=0$ be a smooth plane curve of degree $d\geq4$, and consider the following binary forms: where the coefficients $e_i$ are determined as in eq:A5_coefficients (see Lemma cor:product-minimal). Then, $\operatorname{Aut}_{\operatorname{red}}(\mathcal{X})$ is isomorphic to $\operatorname{A}_5$ if and only if $d=12\epsilon+20\epsilon'+30\epsilon"+60t$, and where $a_i\neq 2^

Theorems & Definitions (67)

  • Theorem 1.1: Case $\operatorname{A}_5$
  • Theorem 1.2: Case $\operatorname{S}_4$
  • Theorem 1.3: Case $\operatorname{A}_{4}$
  • Theorem 1.4: Case $\operatorname{D}_{m}$
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 57 more