{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/curve-classes-on-conic-bundle-threefolds-and","title":"Curve classes on conic bundle threefolds and applications to rationality","arxiv_id":"2207.07093","date":"2022-07-14","proceeding":null,"authors":["Sarah Frei","Lena Ji","Soumya Sankar","Bianca Viray","Isabel Vogt"],"abstract":"We undertake a study of conic bundle threefolds $\\pi\\colon X\\to W$ over geometrically rational surfaces whose associated discriminant covers $\\tilde{\\Delta}\\to\\Delta\\subset W$ are smooth and geometrically irreducible. First, we determine the structure of the group $\\mathrm{CH}^2 X_{\\overline{k}}$ of rational equivalence classes of curves. Precisely, we construct a Galois-equivariant group homomorphism from $\\mathrm{CH}^2X_{\\overline{k}}$ to a group scheme associated to the discriminant cover $\\tilde{\\Delta}\\to \\Delta$ of $X$. The target group scheme is a generalization of the Prym variety of $\\tilde{\\Delta}\\to\\Delta$ and so our result can be viewed as a generalization of Beauville's result that the algebraically trivial curve classes on $X_{\\overline{k}}$ are parametrized by the Prym variety. We apply our structural result on curve classes to study the refined intermediate Jacobian torsor (IJT) obstruction to rationality introduced by Hassett--Tschinkel and Benoist--Wittenberg. The first case of interest is $W = \\mathbb P^2$ and $\\Delta$ is a smooth plane quartic. In this case, we show that the IJT obstruction characterizes rationality when the ground field has less arithmetic complexity (precisely, when the $2$-torsion in the Brauer group of the ground field is trivial). We also show that a hypothesis of this form is necessary by constructing, over any $k \\subset\\mathbb R$, a conic bundle threefold with $\\Delta$ a smooth quartic where the IJT obstruction vanishes, yet $X$ is irrational over $k$.","url_abs":"https://arxiv.org/abs/2207.07093v3","url_pdf":"https://arxiv.org/pdf/2207.07093v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"curve-classes-on-conic-bundle-threefolds-and","repo_url":"https://github.com/ivogt161/fjsvv-rationality","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}