Let $S$ be a smooth projective connected surface over an algebraically closed field $k$ embedded into a projective space $\mathbb{P}^d$ and let $C$ be a smooth projective curve embedded into $S$. Let $\text{CH}_0(S)_{\deg=0}$ and $\text{CH}_0(C)_{\deg=0}$ be the Chow groups of zero cycles of degree $0$ on $S$ and $C$, respectively. Following the approach of Bannerjee and Guletskii we prove that the kernel of the Gysin homomorphism from $\text{CH}_0(C)_{\deg=0}$ to $\text{CH}_0(S)_{\deg=0}$ induced by the embedding is a countable union of translates of an abelian subvariety $A$ inside the Jacobian $J$ of the curve $C$. We also prove that there is a $c$-open subset $U_0$ contained in the set $U \subset (\mathbb{P}^d)^*$ parametrizing the smooth projective curves such that $A=0$ or $A=B$ for all curves parametrized by $U_0$, where $B$ is the abelian subvariety of $J$ corresponding to the vanishing cohomology $H^1(C, k’)_{\text{van}}$ of $C$.

The subset $U_0$ being countable open allows to apply the irreducibility of the monodromy representation on $H^1(C, k’)_{\text{van}}$ (for the étale cohomology and for the singular cohomology for complex algebraic varieties). We describe the Gysin kernel for the points in $U \setminus U_0$ where the local and global monodromy representations are not fully understood. The approach is to construct a stratification $\{U_i \subseteq U \}_{i \in I}$ of $U$ by countable open subsets with $I$ an at most countable, partially ordered set, for each of which the monodromy argument applies. We then apply a convergence argument for the stratification $ \{U_i\}_{i \in I}$ such that the monodromy argument applies for $U$ seen as the set-theoretic directed union $U = \underset{\underset{i}{\rightarrow}}{\cup} \; U_i$.

This is joint work with Rina Paucar Rojas (IMCA/UNI, Lima in Peru).