Fields of moduli and fields of definition of odd signature curves

2011/11/18 by Michela Artebani, Artebani, Michela, Saúl Quispe +1 · 1 citation
Arts and Humanities · Mathematics · #14H10 #14H37 #14H45 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #North African History and Literature

paper · pdf · doi:10.48550/arxiv.1111.4489

openalex publication_date 2011/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a smooth projective algebraic curve of genus g≥ 2 defined over a field K. We show that X can be defined over its field of moduli if it has odd signature, i.e. if the signature of the covering X→ X/\Aut(X) is of type (0;c₁,...,c_k), where some c_i appears an odd number of times. This result is applied to q-gonal curves and to plane quartics. For q-gonal curves, we prove that non-normal q-gonal curves can be defined over their field of moduli and we construct examples of normal q-gonal curves with field of moduli ℝ that can not be defined over ℝ. For plane quartics, we prove that they can be defined over their field of moduli if the automorphism group is not isomorphic to either C₂ or C₂× C₂.

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• Algebraic models of cyclic k-gonal curves

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