On the existence of certain weak Fano threefolds of Picard number two
  This article settles the question of existence of smooth weak Fano threefolds of Picard number two with small anti-canonical map and previously classified numerical invariants obtained by blowing up certain curves on smooth Fano threefolds of Picard number 1 with the exception of 12 numerical cases. 
Maxim Arap, Joseph Cutrone, and Nicholas Marshburn
math.AG