{"paper":{"title":"The field of moduli of plane curves","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AG","authors_text":"Giulio Bresciani","submitted_at":"2023-03-02T18:12:01Z","abstract_excerpt":"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.\n  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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.01454","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2303.01454/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}