@article{5c0f44a887674deab3b521045db512aa,
title = "Curve classes on conic bundle threefolds and applications to rationality",
abstract = "We undertake a study of conic bundle threefolds π: X → W over geometrically rational surfaces whose associated discriminant covers (Formula presented) → Δ (Formula presented) W are smooth and geometrically irreducible. We first show that the structure of the Galois module CH2 Xk of rational equivalence classes of curves is captured by a group scheme that is a generalization of the Prym variety of (Formula presented) → Δ. This generalizes Beauville{\textquoteright}s result that the algebraically trivial curve classes on Xk 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 where W = P2 and Δ 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 (Formula presented) R, a conic bundle threefold with Δ a smooth quartic where the IJT obstruction vanishes, yet X is irrational over k.",
keywords = "Conic bundles, curve classes, intermediate Jacobians, Prym varieties, rationality",
author = "Sarah Frei and Lena Ji and Soumya Sankar and Bianca Viray and Isabel Vogt",
note = "This material is based upon work carried out while the authors attended the 2020 Women in Algebraic Geometry Conference, hosted virtually at the Institute for Computational and Experimental Research in Mathematics in Providence, RI. We thank the organizers of that conference\textbackslash{}u2014Melody Chan, Antonella Grassi, Julie Rana, Rohini Ramadas, and Isabel Vogt\textbackslash{}u2014for providing us the opportunity to work together and thank ICERM for providing the virtual tools to facilitate our collaboration. We also thank Asher Auel, Olivier Benoist, Brendan Hassett, Janos Kollar, Shizhang Li, Bjorn Poonen, Vyacheslav Shokurov, and Olivier Wittenberg for helpful conversations. We especially thank the anonymous referee for a very thorough reading of this paper and helpful comments that have improved both the exposition and results. In particular, we thank the referee for suggesting Example 8.6 and for suggesting a different approach for proving Lemma 5.7, which allowed us to prove the results that now appear in Section 3 in a more widely applicable context. This material is based upon work supported by NSF Grant No. DMS-1439786. During the preparation of this article, S.F. was partially supported by NSF DMS-1745670; L.J. was partially supported by NSF GRFP DGE-1656466, NSF DMS-1840234, and NSF MSPRF DMS-2202444; S.S. was partially supported by NSF DMS-1928930; B.V. was partially supported by NSF DMS-1553459, NSF DMS-2101434, a Simons Fellowship, and the AMS Birman Fellowship; and I.V. was partially supported by NSF MSPRF DMS-1902743 and NSF DMS-2200655. Additionally, this material is based partially upon work that was supported by National Science Foundation grant DMS-1928930 while B.V. and I.V. were in residence at the Simons Laufer Mathematical Sciences Institute in Berkeley, California, during the Spring 2023 semester.",
year = "2024",
doi = "10.14231/AG-2024-014",
language = "English (US)",
volume = "11",
pages = "421--459",
journal = "Algebraic Geometry",
issn = "2313-1691",
publisher = "European Mathematical Society Publishing House",
number = "3",
}