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arXiv · 2303.01454

The field of moduli of plane curves

Giulio Bresciani
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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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Giulio Bresciani. 2023-03-02. The field of moduli of plane curves. https://arxiv.org/abs/2303.01454

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