Cycles over fields of transcendence degree one
  We extend earlier examples provided by Schoen, Nori and Bloch to show that when a surface has the property that the kernel of its Albanese map is non-zero over the field of complex numbers, this kernel is non-zero over a field of transcendence degree one. This says that the conjecture of Bloch and Beilinson that this kernel is zero for varieties over number fields is precise in the sense that it is not valid for fields of transcendence degree one. 
Mark Green, Philip A. Griffiths, and Kapil Hari Paranjape