Rationality problem of three-dimensional monomial group actions
Rationality problem of three-dimensional monomial group actions
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三维单项群行为的理性问题
DOI：
10.1016/j.jalgebra.2011.06.004
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发表时间：
2011
影响因子：
0.9
通讯作者：
山崎 愛一
中科院分区：
文献类型：
--
作者：
星 明考;北山 秀隆;山崎 愛一
设K是特征不为2的域，K（x，y，z）是K上的三元有理函数域。设G是有限群，G通过单名K-自同构作用在K（x，y，z）上.考虑了固定域K（x，y，z）G在G作用下的有理性问题，即K（x，y，z）G是否在K上有理（即纯超越）.我们可以假定G是GL（3，Z）的一个子群，并且这个问题在GL（3，Z）中共轭地确定。在GL（3，Z）中G有73个共轭类.利用Endo-Miyata，Voskresenskietz，Lenstra，Saltman，Hajja，Kang和Yamasaki的结果，证明了GL（3，Z）中2-群的8个共轭类在某个基域K上的某些单项式作用下对该问题有负解，并给出了K（x，y，z）G在K上有理性的充要条件.本文证明了在G的单项式作用下的固定域K（x，y，z）G在K上是有理数的，除了2-群的可能负的8种情形和4次交错群的未知的1种情形.此外，我们给出了K上不动域的显式超越基。对于未知情形，在一定条件下得到了问题的一个肯定解。特别地，我们证明了若K是二次闭域，则K（x，y，z）G是K上的有理域.最后给出了该结果在四维线性Noether问题中的应用.
Let K be a field of characteristic not two and K(x,y,z) the rational function field over K with three variables x,y,z. Let G be a finite group acting on K(x,y,z) by monomial K-automorphisms. We consider the rationality problem of the fixed field K(x,y,z)Gunder the action of G, namely whether K(x,y,z)Gis rational (that is, purely transcendental) over K or not. We may assume that G is a subgroup of GL(3,Z) and the problem is determined up to conjugacy in GL(3,Z). There are 73 conjugacy classes of G in GL(3,Z). By results of Endo–Miyata, Voskresenskiĭ, Lenstra, Saltman, Hajja, Kang and Yamasaki, 8 conjugacy classes of 2-groups in GL(3,Z) have negative answers to the problem under certain monomial actions over some base field K, and the necessary and sufficient condition for the rationality of K(x,y,z)Gover K is given. In this paper, we show that the fixed field K(x,y,z)Gunder monomial action of G is rational over K except for possibly negative 8 cases of 2-groups and unknown one case of the alternating group of degree four. Moreover we give explicit transcendental bases of the fixed fields over K. For the unknown case, we obtain an affirmative solution to the problem under some conditions. In particular, we show that if K is quadratically closed field then K(x,y,z)Gis rational over K. We also give an application of the result to 4-dimensional linear Noetherʼs problem.