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On the field of moduli of superelliptic curves
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On the field of moduli of superelliptic curves
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A superelliptic curve $\X$ of genus $g\geq 2$ is not necessarily defined over its field of moduli but it can be defined over a quadratic extension of it. While a lot of work has been done by many authors to determine which hyperelliptic curves are defined over their field of moduli, less is known for superelliptic curves. In this paper we observe that if the reduced group of a genus $g\geq 2$ superelliptic curve $\X$ is different from the trivial or cyclic group, then $\X$ can be defined over its field of moduli; in the cyclic situation we provide a sufficient condition for this to happen. We also determine those families of superelliptic curves of genus at most $10$ which might not be definable over their field of moduli.
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Cited by 1 Pith paper
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Fields of Moduli of Smooth del Pezzo Surfaces
Every smooth del Pezzo surface of degree at least 3 over an algebraically closed field in characteristic 0 is defined over its field of moduli, but degree-1 and degree-2 counterexamples exist over C/R.
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