Birational maps of del Pezzo fibrations (Q2744704)

Let \({\mathfrak O}\) be a discrete valuation ring with residue field of characteristic zero. And let \(X\) and \(Y\) be del Pezzo fibrations of degree \(d\leq 4\) over \(\text{Spec} {\mathfrak O}\). Suppose that their generic fibers and special fibers are non-singular. In this situation, we show that any birational map between \(X\) and \(Y\) over \(\text{Spec} {\mathfrak O}\) which is biregular on the generic fiber is a biregular morphism. The main method is to apply the minimum of the log canonical thresholds of effective anticanonical divisors on non-singular del Pezzo surfaces for inversion of adjunction.