The kernel of the Gysin homomorphism for Chow groups of zero cycles.

Abstract

Let S be a smooth projective surface over a field k, and let C be a smooth hyperplane section of S. For a closed embedding of S into a projective space P consider the linear system Σ of hyperplane sections and the corresponding discriminant locus ∆ of singular hyperplane sections in the dual space. Let U := Σ \ ∆.

Let CH0(S) and CH0(C) be the Chow groups of 0-cycles of degree 0 of S and C, respectively. We prove that the kernel of the Gysin homomorphism from CH0(C) to CH0(S) induced by the closed embedding of C into S is the countable union of shifts of a certain abelian subvariety A inside J(C), the Jacobian of the curve C. Moreover, for a Zariski countable open subset V in U , for every closed point t in V, either A at t coincides with a certain abelian variety Bt inside J(C), and then the Gysin kernel is a countable union of shifts of Bt, or A at t is 0, in which case the Gysin kernel is countable.

The subset V being countable open allows to apply the irreducibility of the monodromy representation on the vanishing cohomology of a smooth section (for the étale cohomology and for the singular cohomology in a Hodge theoretical context for complex algebraic varieties). We aim to describe the Gysin kernel for the points t in U \ V where the local and global monodromy representations are not fully understood. The approach is to construct a stratification {Ui ⊆ U }i∈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 {Ui}i∈I such that the monodromy argument applies for U seen as the set-theoretic directed union of all Ui.