Zero-cycles on quadric fibrations: Finiteness theorems and the cycle map
Zero-cycles on quadric fibrations: Finiteness theorems and the cycle map
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二次纤维的零周期：有限定理和周期图
DOI：
10.1007/s002220050042
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发表时间：
1996
影响因子：
3.1
通讯作者：
V. Suresh
中科院分区：
文献类型：
--
作者：
R. Parimala;V. Suresh
由于 Merkurjev 和 Suslin 的深入研究结果（参见[CT2]），对于局部和全局域上的变体，余维 2 中循环的 Chow 群的挠子群的结构已经相当容易理解。然而，对于维数大于 2 的簇 X 上 0 循环的 Chow 群 CH0 (X) 中的挠率知之甚少。例如，尚不清楚 CH0 (X) 是否是数域上的平滑射影簇的有限生成。在本文中，我们研究了曲线上二次纤维振动的 0 循环 Chow 群。我们的出发点是 Colliot-Thdl6ne 和 Skorobogatov [CTS] 的一篇精美论文，其中识别了曲线上二次纤维振动上的“相对”Chow 群 0 周期，以及曲线函数域的非零元素组的某个子商，根据定义通用纤维的二次形式的值来定义。令 rc'X-~C 为相对维数 d > 1 的“容许二次纤维化”（参见第 0 节），是特征场 k 不为 2 的平滑投影曲线 C，并令 CHo (X/C) 表示同态 re 的核，9 CH0 (X)~ CH0 (C)。利用上述相对 Chow 群 CHo (X/C) 的表征，Colliot-Th616ne 和 Skorobogatov 证明了当 d 5 2 和 k 是数域（除本文中处理的一种特殊情况）或局部域时 CHo (X/C)[CTS，定理 6.2] 的有限性。他们还表明，如果 k 是上同调维数为 1 的域，则 CHo (X/C) 为零。他们在更一般的情况下提出了以下问题。 i) 如果 k 是有理数域 Q 上的有限生成域，那么群 CHo (X/C) 是有限的吗？ ii) 如果 d> 3 且上同调维数 ka 域小于或等于 2，则群 CHo (X/C) 为零，或者至少是有限的？
The structure of the torsion subgroup of the Chow group of cycles in codimension 2, for varieties over local and global fields is fairly well understood, thanks to the deep results of Merkurjev and Suslin (cf.[CT2]). However, much less is known on the torsion in the Chow group CH0 (X) of 0-cycles on varieties X of dimension greater than 2. For instance, it is not known whether CH0 (X) is finitely generated for smooth projective varieties over number fields. In this paper, we study the Chow group of 0-cycles on quadric fibrations over curves. Our starting point is an elegant paper of Colliot-Thdl6ne and Skorobogatov [CTS], in which there is an identification of the" relative" Chow group of 0-cycles on a quadric fibration over a curve, with a certain subquotient of the group of nonzero elements of the function field of the curve, defined in terms of the values of the quadratic form defining the generic fibre. Let rc'X-~ C be an" admissible quadric fibration"(cf. Sect. 0) of relative dimension d> 1, of a smooth projective curve C over a field k of characteristic not 2 and let CHo (X/C) denote the kernel of the homomorphism re, 9 CH0 (X)~ CH0 (C). Using the above characterisation of the relative Chow group CHo (X/C), Colliot-Th616ne and Skorobogatov proved finiteness of CHo (X/C)[CTS, Theorem 6.2] when d 5 2 and k is a number field (except in one special situation, which is treated in this paper) or a local field. They also show that CHo (X/C) is zero if k is a field of cohomological dimension 1. They raised the following questions in a more general setting. i) If k is a finitely generated field over the field Q of rational numbers, is the group CHo (X/C) finite? ii) If d> 3 and ka field of cohomological dimension less than or equal to 2, is the group CHo (X/C) zero, or is it at least finite'?