AI 中文总结
研究有限域上直线定义的圆锥束合理性，固定退化纤维数\(n\)，通过伽罗瓦作用经外尔群\(W(D_n)\)将合理性归为带符号置换类型条件，利用相关工作及切博塔廖夫密度计算\(q\)趋于无穷时有理圆锥束的渐近比例。
AI 中文摘要
我们研究在直线上定义的有限域上圆锥束的合理性。特别地，对于固定数量\(n\)的退化纤维，在阶为\(q\)的有限域上为有理的好圆锥束的比例是多少？其皮卡群上的伽罗瓦作用通过外尔群\(W(D_n)\)分解，将合理性归结为关于带符号置换类型的条件。利用科利奥 - 泰莱纳和杨的工作以及切博塔廖夫密度，我们计算了随着\(q\)趋于无穷时此类有理圆锥束的渐近比例。
英文摘要
We study rationality of conic bundles over finite fields defined over the line. In particular, for a fixed number $n$ of degenerate fibers, what is the proportion of good conic bundles that are rational over a finite field of order $q$? The Galois action on their Picard groups factors through the Weyl group $W(D_n)$, reducing rationality to a condition on signed permutation types. Using work of Colliot-Thélène and Yang together with Chebotarev density, we compute the asymptotic proportion of such conic bundles that are rational as $q$ goes to infinity.
Comments19 pages, 1 figure
