By Jahnke–Peternell–Radloff and Takeuchi, almost Fano threefolds with del Pezzo fibrations were classified. Among them, there exist 10 classes such that the existence of members of these was not proved. In this paper, we construct such examples belonging to each of 10 classes.

We classify rank two vector bundles on a given del Pezzo threefold of degree four whose projectivizations are weak Fano into seven cases. We also give an example for each of these seven cases.

We classify rank two vector bundles on a given del Pezzo threefold of degree four whose projectivizations are weak Fano into seven cases. We also give an example for each of these seven cases.

In this paper, we study a sextic del Pezzo fibration over a curve comprehensively. We obtain certain formulae of several basic invariants of such a fibration. We also establish the embedding theorem of such a fibration which asserts that every such a fibration is a relative linear section of a Mori fiber space with the general fiber (P 1 ) 3 and that with the general fiber (P 2 ) 2 . As an application of this embedding theorem, we classify singular fibers of such a fibrations and answer a question of T. Fujita about existence of non-normal fibers.

We classify rank two vector bundles on a del Pezzo threefold X of Picard rank one whose projectivizations are weak Fano. We also investigate the moduli spaces of such vector bundles when X is of degree five, especially whether it is smooth, irreducible, or fine.