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The field of moduli of plane curves
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abstract
We prove that a smooth, complex plane curve of odd degree can be defined by a polynomial with coefficients in $\mathbb{R}$ if and only if it is isomorphic to its complex conjugate; there are counterexamples in even degree. Over arbitrary base fields of characteristic $0$, we prove that a smooth plane curve of degree prime with $6$ can be defined by a polynomial with coefficients in the field of moduli. We also prove results about fields of moduli of algebraic cycles in $\mathbb{P}^{2}$. In particular, these apply to singular plane curves of arbitrary degree, too.
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Cited by 1 Pith paper
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On the Fields of Moduli of Curves of Genus Six
For genus-6 curves, bielliptic curves descend to their field of moduli, and non-descent can only occur for curves on smooth degree-5 del Pezzo surfaces with automorphism group C2 or D10 (with √-1 outside the base fiel...
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