The variation of the Gysin kernel in a family

TL;DR

This paper studies how the kernels of push-forward maps from Jacobians of fibers to the surface's zero-cycle group vary in a family, showing they form a structured family of abelian varieties over the base curve.

Contribution

It demonstrates that the kernels of the push-forward homomorphisms form a countable union of translates of an abelian scheme over the base curve.

Findings

The kernels form a family parametrized by the base curve.

The family is a countable union of translates of an abelian scheme.

The structure of the kernels varies in a controlled, algebraic manner.

Abstract

Consider a smooth projective surface $S$. Consider a fibration $S\to C$ where $C$ is a quasi-projective curve such the fibers are smooth projective curves. The aim of this text is to show that the kernels of the push-forward homomorphism $\{j_{t*}\}_{t\in C}$ from the Jacobian $J(C_t)$ to $A_0(S)$ forms a family in the sense that it is a countable union of translates of an abelian scheme over $C$ sitting inside the Jacobian scheme $\mathscr{J}\to C$, such that the fiber of this countable union at $t$ is the kernel of $j_{t*}$.