Finiteness theorems for algebraic cycles of small codimension on quadric   fibrations over curves
  We obtain finiteness theorems for algebraic cycles of small codimension on quadric fibrations X over curves over perfect fields k. For example, if k is finitely generated over Q and the fibration has odd relative dimension at least 11, then CH^{i}(X) is finitely generated for i<=4. 
Cristian D. Gonz\'alez-Avil\'es