Cycles on curves over global fields of positive characteristic

Let $k$ be a global field of positive characteristic, and let $\sigma: X \longrightarrow \operatorname{Spec} k$ be a smooth projective curve. We study the zero-dimensional cycle group $V(X) =\operatorname{Ker}(\sigma_*: SK_1(X) \rightarrow K_1(k))$ and the one-dimensional cycle group $W(X) =\operatorname{coker}(\sigma^*: K_2(k) \rightarrow H^0_{Zar}(X, \mathcal{K}_2))$, addressing the conjecture that $V(X)$ is torsion and $W(X)$ 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 $K$-groups of a certain smooth projective surface over a finite field.