Birational Maps of Del Pezzo Fibrations

In classical result, it is known that any P-bundle over a nonsingular curve T can be birationally transformed to P-bundle over a nonsingular curve T by an elementary transformation. Here, we can ask if it is also possible in 3-fold case. In other words, is it true that any nonsingular del Pezzo fibration over a nonsingular curve can be transformed to another nonsingular del Pezzo fibration? In this question, we can add more condition on del Pezzo fibrations with some kind of analogue from ruled surface cases, that is, we can assume that their fibers are always nonsingular even though this is not true for any nonsingular del Pezzo fibration. Now, we ask the same question for local cases. Of course, we can birationally transform any P-bundle over a germ of nonsingular curve (T, o) into another P-bundle over (T, o). But, in del Pezzo fibrations over (T, o), something different happens. In this paper, we will show that any del Pezzo fibration of degree d ≤ 4 with nonsingular special fiber cannot be birationally transformed into another del Pezzo fibration with nonsingular special fiber. Let O be a discrete valuation ring such that its residue field k is of characteristic zero. We denote K the quotient field of O.