Parameter spaces for algebraic equivalence

TL;DR

This paper extends classical results on algebraic triviality of cycles, showing that over any base field, it suffices to consider families parameterized by curves or abelian varieties, which aids in broader algebraic geometry applications.

Contribution

It generalizes Weil's results on algebraic triviality of cycles from algebraically closed fields to arbitrary base fields, enhancing the scope of algebraic equivalence theory.

Findings

Extension of Weil's results to arbitrary fields

Reduction to families over curves or abelian varieties

Application to algebraic representatives over perfect fields

Abstract

A cycle is algebraically trivial if it can be exhibited as the difference of two fibers in a family of cycles parameterized by a smooth scheme. Over an algebraically closed field, it is a result of Weil that it suffices to consider families of cycles parameterized by curves, or by abelian varieties. In this paper, we extend these results to arbitrary base fields. The strengthening of these results turns out to be a key step in our work elsewhere extending Murre's results on algebraic representatives for varieties over algebraically closed fields to arbitrary perfect fields.