On Del Pezzo fibrations over curves (Q752119)

The author studies Del Pezzo fibrations over curves, i.e. proper surjective holomorphic mappings f: \(M\to C\) with \(\dim (C)=1\), \(\dim (M)=n+1\geq 3\) together with an f-ample line bundle \({\mathcal L}\) on M such that for a general point \(x\in C\), \(M_ x=f^{-1}(x)\) is a Del Pezzo manifold, i.e. \(K+(n-1){\mathcal L}=0\) in \(Pic(M_ x)\), where K is the canonical bundle of M. These fibrations generalize elliptic surfaces in the surface theory. The author classifies reducible fibers (there exist eight types of degenerations for \(n=2\) and one type for \(n=3)\) and gives necessary conditions for existence of non-normal fibers (they appear very rarely if ever). Finally, he studies possible global structures. The paper heavily relies on the preceding papers by the author.