Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algebraic models of cyclic $k$-gonal curves

Ruben A. Hidalgo, Sebastián Reyes-Carocca

TL;DR

The paper develops explicit algebraic models for tame cyclic k-gonal curves by describing the curve S via a Galois cover with deck group <tau>, and by encoding the action of the normalizer N through its quotient barN in PSL2(C). It systematizes the possible quotient signatures S/N according to the finite subgroups barN (cyclic, dihedral, A4, S4, A5), and provides concrete algebraic descriptions of S in terms of barN, including detailed cases when barN contains Z_m, D_m, A4, S4, or A5, with formulas for the lifted automorphisms and the polynomials f(x). The work extends prior prime-k results (notably Wootton) to arbitrary k by presenting an executable algebraic framework and offering explicit examples for small n (2–5), thereby enabling the explicit construction of tamely cyclic k-gonal curves with prescribed symmetries. The methods are algebraic, characteristic-free (when k is not divisible by the characteristic), and suitable for computation, contributing to the understanding of automorphism groups of cyclic gonal curves and their moduli. Overall, the paper provides a comprehensive toolkit for writing down explicit equations reflecting N-action across a full spectrum of possible normalizers, with extensive case studies and prime-situation generalizations.

Abstract

In this paper, we describe explicit algebraic equations of tame cyclic $k$-gonal curves, where $k \geq 2$ is an integer, reflecting the action of the normalizer of a tame cyclic $k$-gonal automorphism. For $k$ a prime integer, this was previously done by A. Wootton.

Algebraic models of cyclic $k$-gonal curves

TL;DR

The paper develops explicit algebraic models for tame cyclic k-gonal curves by describing the curve S via a Galois cover with deck group <tau>, and by encoding the action of the normalizer N through its quotient barN in PSL2(C). It systematizes the possible quotient signatures S/N according to the finite subgroups barN (cyclic, dihedral, A4, S4, A5), and provides concrete algebraic descriptions of S in terms of barN, including detailed cases when barN contains Z_m, D_m, A4, S4, or A5, with formulas for the lifted automorphisms and the polynomials f(x). The work extends prior prime-k results (notably Wootton) to arbitrary k by presenting an executable algebraic framework and offering explicit examples for small n (2–5), thereby enabling the explicit construction of tamely cyclic k-gonal curves with prescribed symmetries. The methods are algebraic, characteristic-free (when k is not divisible by the characteristic), and suitable for computation, contributing to the understanding of automorphism groups of cyclic gonal curves and their moduli. Overall, the paper provides a comprehensive toolkit for writing down explicit equations reflecting N-action across a full spectrum of possible normalizers, with extensive case studies and prime-situation generalizations.

Abstract

In this paper, we describe explicit algebraic equations of tame cyclic -gonal curves, where is an integer, reflecting the action of the normalizer of a tame cyclic -gonal automorphism. For a prime integer, this was previously done by A. Wootton.

Paper Structure

This paper contains 32 sections, 5 theorems, 95 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algebraic model
  4. The normalizer subgroup of $\langle \tau \rangle$
  5. The possibilities for the signatures of $S/N$
  6. The cyclic case
  7. The dihedral case
  8. The alternating case ${\mathcal{A}}_{4}$
  9. The symmetric case ${\mathfrak S}_{4}$
  10. The alternating case ${\mathcal{A}}_{5}$
  11. The algebraic descriptions of $S$ in terms of $\bar{N}$: case $\tau$ tame
  12. A computational remark
  13. The case when $\bar{N}$ contains ${\mathbb Z}_{m}$
  14. The case $s_{0}=0$
  15. The case $s_{0}=1$
  16. ...and 17 more sections

Key Result

Lemma 1

Let $\eta \in N$ and assume $\tau$ is tame. If $\theta(\eta)$ fixes a point in ${\mathcal{B}}$, then $\eta$ commutes with $\tau$, that is, $u_{\eta}=1$.

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Lemma 2: Lemma 1 in HQS
  • Corollary 1: HQS
  • Lemma 3
  • proof
  • Lemma 4
  • proof