Halphen Pencils on Weighted Fano Threefold Hypersurfaces
  We classify all pencils on a general weighted hypersurface in $\mathbb{P}(1,a_{1},a_{2},a_{3},a_{4})$ of degree $\sum_{i=1}^{4}a_{i}$ whose general members are surfaces of Kodaira dimension zero. 
Ivan Cheltsov and Jihun Park