{
  "id": "2309.12192",
  "version": "v1",
  "published": "2023-09-21T15:58:53.000Z",
  "updated": "2023-09-21T15:58:53.000Z",
  "title": "Real versus complex plane curves",
  "authors": [
    "Giulio Bresciani"
  ],
  "comment": "Comments are welcome! In particular, I would be very grateful for historical information regarding the statement of Theorem 1",
  "categories": [
    "math.AG",
    "math.CV"
  ],
  "abstract": "We prove that a smooth, complex plane curve $C$ of odd degree can be defined by a polynomial with real coefficients if and only if $C$ is isomorphic to its complex conjugate. Counterexamples are known for curves of even degree. More generally, we prove that a plane curve $C$ over an algebraically closed field $K$ of characteristic $0$ with field of moduli $k_{C}\\subset K$ is defined by a polynomial with coefficients in $k'$, where $k'/k_{C}$ is an extension with $[k':k_{C}]\\le 3$ and $[k':k_{C}]\\mid \\operatorname{deg} C$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-21T15:58:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "complex plane curve",
      "real coefficients",
      "complex conjugate",
      "odd degree",
      "polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}