Cycles on curves over global fields of positive characteristic

Let be a global field of positive characteristic, and let be a smooth projective curve. We study the zero-dimensional cycle group and the one-dimensional cycle group , addressing the conjecture that is torsion and is finitely generated. The main idea is to use Abhyankar's Theorem on resolution of singularities to relate the study of these cycle groups to that of the -groups of a certain smooth projective surface over a finite field.