Birational classification of fields of invariants for groups of order 128
Birational classification of fields of invariants for groups of order 128
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128 阶群不变量域的双有理分类
DOI：
10.1016/j.jalgebra.2015.05.035
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发表时间：
2016
影响因子：
0.9
通讯作者：
Akinari Hoshi
中科院分区：
文献类型：
--
作者：
Akinari Hoshi;Aiichi Yamasaki;Akinari Hoshi
设G是作用在有理函数域C（xg：g∈ G）上的有限群，对任意g，h∈ G，C-自同构h（xg）= xhg. Noether问题是问不变域C（G）= k（x g：g∈ G）G在C上是否有理（即纯超越）。根据Fischer定理，当G是有限交换群时，C（G）是C上的有理群。Saltman和Bogomolov分别证明了对于任何素数p，存在p阶9和p阶6的群G，使得C（G）在C上不是有理数，这是通过证明非分歧的Brauer群的非零性来实现的：Br nr（C（G））≠ 0，它是双有理不变量H3（X，其中X是函数域为C（G）的光滑射影复簇。对于p= 2，Chu，Hu，Kang和Prokhorov证明了：如果G是阶≤ 32的2-群，则C（G）是C上的有理群。Chu，Hu，Kang和Kunyavskii证明了：如果G是64阶群，则C（G）在C上是有理数的，除了群G属于两个等倾群族Φ 13和Φ 16，其中Br nr（C（G））= 0和Br nr（C（G））<$C2. Bogomolov和Böhning定理指出，如果G1和G2属于同一个等倾族，则C（G1）和C（G2）是稳定C-同构的。研究了128阶群G的C（G）的双有理分类，其中Brnr（C（G））≥ 0. Moravec证明了存在220个128阶群G，Br nr（C（G））= 0，形成11个等倾群族Φ j.我们证明了如果G1和G2分别属于Φ 16，Φ 31，Φ 37，Φ 39，Φ 43，Φ 58，Φ 60或Φ 80（分别为Φ 16，Φ 31，Φ 37，Φ 39，Φ 43，Φ 58，Φ 60或Φ 80），则群G的等倾群族Φ j是有限的.Φ 106或Φ 114），则C（G1）和C（G2）稳定C-同构，Brnr（C（G i））C2。给出了非有理域C（G）的各种情形的显式结构，包括Φ 30且Brnr（C（G））<$C2 × C2的情形。
Let G be a finite group acting on the rational function field C (x g: g∈ G) by C-automorphisms h (x g)= x h g for any g, h∈ G. Noether's problem asks whether the invariant field C (G)= k (x g: g∈ G) G is rational (ie purely transcendental) over C. By Fischer's theorem, C (G) is rational over C when G is a finite abelian group. Saltman and Bogomolov, respectively, showed that for any prime p there exist groups G of order p 9 and of order p 6 such that C (G) is not rational over C by showing the non-vanishing of the unramified Brauer group: Br nr (C (G))≠ 0, which is an avatar of the birational invariant H 3 (X, Z) tors given by Artin and Mumford where X is a smooth projective complex variety whose function field is C (G). For p= 2, Chu, Hu, Kang and Prokhorov proved that if G is a 2-group of order≤ 32, then C (G) is rational over C. Chu, Hu, Kang and Kunyavskii showed that if G is of order 64, then C (G) is rational over C except for the groups G belonging to the two isoclinism families Φ 13 with Br nr (C (G))= 0 and Φ 16 with Br nr (C (G))≃ C 2. Bogomolov and Böhning's theorem claims that if G 1 and G 2 belong to the same isoclinism family, then C (G 1) and C (G 2) are stably C-isomorphic. We investigate the birational classification of C (G) for groups G of order 128 with Br nr (C (G))≠ 0. Moravec showed that there exist exactly 220 groups G of order 128 with Br nr (C (G))≠ 0 forming 11 isoclinism families Φ j. We show that if G 1 and G 2 belong to Φ 16, Φ 31, Φ 37, Φ 39, Φ 43, Φ 58, Φ 60 or Φ 80 (resp. Φ 106 or Φ 114), then C (G 1) and C (G 2) are stably C-isomorphic with Br nr (C (G i))≃ C 2. Explicit structures of non-rational fields C (G) are given for each cases including also the case Φ 30 with Br nr (C (G))≃ C 2× C 2.