2018/01/20 by Akinari Hoshi, Hoshi, Akinari, Ming-chang Kang +5
Mathematics · #12F20 #13A50 #14E08 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:12F20 #msc:13A50 #msc:14E08
paper · pdf · doi:10.48550/arxiv.1801.06616
To appear in Manuscripta Math. The main theorems (old Theorem 1.7 and Theorem 1.8) incorporated into (new) Theorem 1.8. Section 3 and Section 4 interchanged
arxiv created 2021/05/09 · arxiv updated 2021/05/11
Let k be a field with char k≠ 2 and k be not algebraically closed. Let a∈ k∖ k2 and L=k(√(a))(x,y) be a field extension of k where x,y are algebraically independent over k. Assume that σ is a k-automorphism on L defined by σ: √(a)↦ -√(a), x↦ (b)/(x), y↦ (c(x+(b)/(x))+d)/(y) where b,c,d ∈ k, b≠ 0 and at least one of c,d is non-zero. Let L⟨σ⟩=\u∈ L:σ(u)=u\ be the fixed subfield of L. We show that L⟨σ⟩ is isomorphic to the function field of a certain surface in P4k which is given as the intersection of two quadrics. We give criteria for the k-rationality of L⟨σ⟩ by using the Hilbert symbol. As an appendix of the paper, we also give an alternative geometric proof of a part of the result which is provided to the authors by J.-L. Colliot-Thélène.