2016/09/30 by Rubén A. Hidalgo, Hidalgo, Ruben A., Saúl Quispe +3 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1609.09576
openalex publication_date 2016/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A closed Riemann surface \mathcal X, of genus g ≥ 2, is called a generalized superelliptic curve of level n ≥ 2 if it admits an order n conformal automorphism τ so that \mathcal X/⟨ τ⟩ has genus zero and τ is central in \rm Aut(\mathcal X); the cyclic group H=⟨ τ⟩ is called a generalized superelliptic group of level n for \mathcal X. These Riemann surfaces are natural generalizations of hyperelliptic Riemann surfaces (when n=2). We provide an algebraic curve description of these Riemann surfaces in terms of their groups of automorphisms. Also, we observe that the generalized superelliptic group H of level n is unique, with the exception of a very particular family of exceptional generalized superelliptic Riemann surfaces for n even. In particular, the uniqueness holds if either: (i) n is odd or (ii) the quotient \mathcal X/H has all its cone points of order n (for instance, when \mathcal X is a superelliptic curve of level n). In the non-exceptional case, we use this uniqueness property of its generalized superelliptic group H to observe that the corresponding curves are definable over their fields of moduli if \rm Aut(\mathcal X)/H is neither trivial or cyclic.