We describe the Griffiths group of the product of a curve and a surface as a quotient of the Albanese kernel of over the function field of . When is a hyperplane section of varying in a Lefschetz pencil, we prove the nonvanishing in Griff of a modification of the graph of the embedding for infinitely many members of the pencil, provided the ground field is of characteristic , the geometric genus of is , and is large or is “of motivated abelian type”.
Bruno Kahn. "Albanese kernels and Griffiths groups." Tunisian J. Math. 3 (3) 589 - 656, 2021. https://doi.org/10.2140/tunis.2021.3.589