论文标题
真实的圆锥捆的合理性与四分之一的判别曲线
Rationality of real conic bundles with quartic discriminant curve
论文作者
论文摘要
我们研究了$ \ Mathbb P^1 \ times \ Mathbb P^2 $的真实双重盖，在$（2,2）$ - divisor上分支，通过第二个预测，它们具有三倍的圆锥束的结构，具有平滑的四倍歧视曲线。在平滑平面四分之一的每种同位素类别中，我们构造了圆锥束的总空间是合理的示例。对于六个同位素类别中的五个，我们构建了$ \ Mathbb C $ - 理性的示例，这些示例对$ \ Mathbb r $具有理性的障碍，而对于第六类，我们表明我们认为的模型都是合理的。此外，对于具有非理性成员的五个类别中的三个，我们使用真实基因座的拓扑以及Hassett-tschinkel和Benoist的中间Jacobian Torsor障碍物（Wittenberg）提供了理性的特征。我们考虑的双盖模型是由S. Frei，S。Sankar，B。Viray，I。Vogt和第一作者引入和研究的。
We study real double covers of $\mathbb P^1\times\mathbb P^2$ branched over a $(2,2)$-divisor, which have the structure of a conic bundle threefold with smooth quartic discriminant curve via the second projection. In each isotopy class of smooth plane quartics, we construct examples where the total space of the conic bundle is rational. For five of the six isotopy classes we construct $\mathbb C$-rational examples that have obstructions to rationality over $\mathbb R$, and for the sixth class, we show that the models we consider are all rational. Moreover, for three of the five classes with irrational members, we give characterizations of rationality using the topology of the real locus and the intermediate Jacobian torsor obstruction of Hassett--Tschinkel and Benoist--Wittenberg. The double cover models we consider were introduced and previously studied by S. Frei, S. Sankar, B. Viray, I. Vogt, and the first author.
