Curve classes on conic bundle threefolds and applications to rationality
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866916432536666112 |
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| author | Frei, Sarah Ji, Lena Sankar, Soumya Viray, Bianca Vogt, Isabel |
| author_facet | Frei, Sarah Ji, Lena Sankar, Soumya Viray, Bianca Vogt, Isabel |
| contents | We undertake a study of conic bundle threefolds $π\colon X\to W$ over geometrically rational surfaces whose associated discriminant covers $\tildeΔ\toΔ\subset W$ 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 $\tildeΔ\to Δ$ of $X$. The target group scheme is a generalization of the Prym variety of $\tildeΔ\toΔ$ 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 $Δ$ 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 $k \subset\mathbb R$, a conic bundle threefold with $Δ$ a smooth quartic where the IJT obstruction vanishes, yet $X$ is irrational over $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_07093 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Curve classes on conic bundle threefolds and applications to rationality Frei, Sarah Ji, Lena Sankar, Soumya Viray, Bianca Vogt, Isabel Algebraic Geometry 14C25 (Primary), 14E08, 14G27, 14H40, 14K30 (Secondary) We undertake a study of conic bundle threefolds $π\colon X\to W$ over geometrically rational surfaces whose associated discriminant covers $\tildeΔ\toΔ\subset W$ 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 $\tildeΔ\to Δ$ of $X$. The target group scheme is a generalization of the Prym variety of $\tildeΔ\toΔ$ 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 $Δ$ 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 $k \subset\mathbb R$, a conic bundle threefold with $Δ$ a smooth quartic where the IJT obstruction vanishes, yet $X$ is irrational over $k$. |
| title | Curve classes on conic bundle threefolds and applications to rationality |
| topic | Algebraic Geometry 14C25 (Primary), 14E08, 14G27, 14H40, 14K30 (Secondary) |
| url | https://arxiv.org/abs/2207.07093 |