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By Jahnke-Peternell-Radloff and Takeuchi, almost Fano threefolds with del Pezzo fibrations were classified. Among them, there exists 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 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.
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 provide a semiorthogonal decomposition for the derived category of fibrations of quintic del Pezzo surfaces with rational Gorenstein singularities. There are three components, two of which are equivalent to the derived categories of the base and t
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
We construct a Kawamata type semiorthogondal decomposition for the bounded derived category of coherent sheaves of nodal quintic del Pezzo threefolds.