A two-dimensional rationality problem and intersections of two quadrics

Let k be a field with char k ≠ 2 and k be not algebraically closed. Let a ∈ k \ k 2 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 P k 4 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.