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.
From: Giulio Bresciani [view email]
[v1]
Thu, 2 Mar 2023 18:12:01 UTC (23 KB)
[v2]
Fri, 3 Mar 2023 14:31:00 UTC (24 KB)
[v3]
Sun, 23 Jul 2023 17:27:05 UTC (27 KB)
[v4]
Wed, 17 Jun 2026 09:37:44 UTC (32 KB)
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