Factorization of birational morphisms with fiber dimension bounded by 1 (Q1119976)

In this note the authors give, both in the local and in the global case, an algebraic proof of the following statement: Let \(f: X\to Y\) be a projective birational morphism of regular noetherian integral schemes with fibres of dimension \(\leq 1\), then f is a composition of blow-ups with non singular centres of codimension 2. This result, which includes the Zariski theorem for birational morphisms of smooth algebraic surfaces, extends to schemes a similar result for smooth algebraic varieties proved by \textit{V. I. Danilov} [Math. USSR, Izv. 16, 419-429 (1981); translation of Izv. Akad. Nauk SSSR, Ser. Mat. 44, 465-477 (1980; Zbl 0453.14004)].