A note on the birational cancellation problem (Q1184031)

Let \(V_ 1\) and \(V_ 2\) be irreducible algebraic varieties over a field \(k\) and let \(f:V_ 1\times\mathbb{P}^ n(k)\to V_ 2\times\mathbb{P}^ n(k)\) be a birational map. This note is dealing with the problem to find conditions in order that \(V_ 1\) and \(V_ 2\) have to be birationally equivalent. --- It contains a useful survey of the classical results on this subject and it extends two theorems of Deveney on ruled function fields to the case \(n\geq 2\).