Albanese Kernels and Griffiths Groups

We describe the Griffiths group of the product of a curve C and a surface S as a quotient of the Albanese kernel of S over the function field of C. When C is a hyperplane section of S varying in a Lefschetz pencil, we prove the nonvanishing in Griff(C x S) of a modification of the graph of the embedding C hooked right arrow S for infinitely many members of the pencil, provided the ground field k is of characteristic 0, the geometric genus of S is > 0, and k is large or S is "of motivated abelian type".