The kernel of the Gysin homomorphism for smooth projective curves

Let $S$ be a smooth projective connected surface over an algebraically closed field $k$ and $\mathrm{\Sigma }$ the linear system of a very ample divisor $D$ on $S$. Let $d:=\dim (\mathrm{\Sigma })$ be the dimension of $\mathrm{\Sigma }$ and ${\phi }_{\mathrm{\Sigma }}:S\hookrightarrow {\mathbb{P}}^d$ the closed embedding of $S$ into ${\mathbb{P}}^d$, induced by $\mathrm{\Sigma }$. For any closed point $t\in \mathrm{\Sigma }\cong {\mathbb{P}}^{d^{\ast }}$, let $C_t$ be the corresponding hyperplane section on $S$, and let $r_t:C_t\hookrightarrow S$ be the closed embedding of the curve $C_t$ into $S$. Let $\mathrm{\Delta }:=\{t\in \mathrm{\Sigma }:C_t\text{ is singular}\}$ be the discriminant locus of $\mathrm{\Sigma }$ and let $U:=\mathrm{\Sigma }\setminus \mathrm{\Delta }$. For $t\in U$, the kernel of the Gysin homomorphism of the Chow groups of $0$-cycles of degree zero, from $C{H}_0(C_t{)}_{deg=0}$ to $C{H}_0(S{)}_{deg=0}$ is the countable union of shifts of a certain abelian subvariety $A_t$ inside $J(C_t)$, the Jacobian of the curve $C_t$ (\cite{PS24} for $k\cong \mathbb{C}$, \cite{SW25} for $k\cong \overline{{\mathbb{F}}_q((t))}$). We prove that for every closed point $t\in U$ either $A_t$ coincides with the abelian variety $B_t$ inside $J(C_t)$ corresponding to the vanishing cohomology $H^1(C_t,k^{\mathrm{\prime }}{)}_{van}$, where $k^{\mathrm{\prime }}$ is the minimal field of definition of $k$, and then the Gysin kernel is a countable union of shifts of $B_t$, or $A_t=0$, in which case the Gysin kernel is countable. Using the language of algebraic stacks as a generalisation of algebraic varieties this is done by constructing an increasing filtration of Zariski countable open substacks $U_i,i\in I,$ of $U$, where $I$ is a countable set and by applying a convergence argument.