Arithmetic of Gysin kernels

Let $k$ be a field, and let $X$ be a smooth projective surface over $k$.

Fix a Lefschetz pencil on $X$, and let $C$ be its fibre at the generic point of $\mathbb P^1$.

The closed immersion of $C$ in to $X_{k(t)}$ induces the Gysin homomorphism from the Jacobian $A$ of the curve $C$ to the Chow group $A_0(X_{k(t)})$ of $0$-cycles of degree $0$ on $X_{k(t)}$.

Embedding $k(t)$ in to an uncountable universal domain $\Bbbk $, we obtain the corresponding homomorphism from $A(\Bbbk )$ to $A_0(X_{\Bbbk })$, whose kernel is either countable or the union of translates of a certain abelian subvariety inside $A_{\Bbbk }$, due to the Deligne-Katz irreducibility of monodromy action on vanishing cycles.

We prove three dichotomy theorems on the structure of the kernel of Gysin homomorphism on $0$-cycles, in terms of étale monodromy action, and working over a field of arbitrary characteristic.