Curve classes on conic bundle threefolds and applications to rationality
| Autor: | Frei, Sarah, Ji, Lena, Sankar, Soumya, Viray, Bianca, Vogt, Isabel |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Algebr. Geom. 11 (3) (2024) 421-459 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.14231/AG-2024-014 |
| Popis: | 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 $k$. Comment: 39 pages. Comments welcome! v2: Updated introduction. v3: Added Section 3, Subsection 5.5, and Example 8.6 |
| Databáze: | arXiv |
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