Note on Conic Bundles over Henselian Discrete Valued Fields with Real Closed Residue Field

Abstract

Let K be the fraction field of a Henselian discrete valuation ring. In Bazyleu et al. (2007) an explicit description of the central simple algebras of exponent 2 in the kernel of the natural map 2 Br(K(x)) → ∏ω∈Ω 2 Br(K(x)ω), (the product taken over all orderings ω and K(x)ω the real closure of K(x) at ω), is given. This allows to describe a class of conic bundle surfaces over K in terms of their local data (i.e., in terms of their degenerate fibres). Such a description is given in Theorem 3.1, the main result of this note.