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We refine the classification of weak Fano threefolds with sextic del Pezzofibrations by considering the Hodge numbers of them. By the refinedclassification result, such threefolds are classified into 17 cases. The mainresult of this paper is to show that there exists an example for each of 17cases, except for 2 cases.

Refinement of the classification of weak Fano threefolds with sextic del Pezzo fibrations