On the existence of almost Fano threefolds with del Pezzo fibrations
  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. 
Takeru Fukuoka
math.AG