Birational maps of standard projective plane bundles over algebraic surfaces

Let X be a smooth algebraic surface with the function field K and let τ: V → X be a standard P^2-bundle over X, i.e. τ is a flat contraction morphism of an extremal ray of a smooth projective variety V with the generic fibre isomorphic to a K-form of P^2, i.e. V ×x K^^- = P^2 for the algebraic closure K^^- of K. In this paper, some birational maps from V to a standard P^2-bundle W are represented by compositions of elementary birational morphisms, where W is a standrd P^2-bundle over the blow-up of X at a point of the non-smooth locus △ of τ. Let C be a smooth curve on X intersecting △ transeversely at one point. A birational map from V to a standard P^2-bundle over X which is isomorphic over X - C, is decomposed into elementary birational morphisms. These are generalizations of the results about standrd conic bundles by V. G. Sarkisov (Math. USSR. Izv. 20)