Some elementary remarks on lci algebraic cycles

TL;DR

This paper reviews known results on representing algebraic cycles with 'nice' cycles like smooth or local complete intersection cycles, and offers new elementary observations on these representation problems.

Contribution

It compiles existing results and introduces new elementary insights on representing algebraic cycles by well-behaved cycles up to rational equivalence.

Findings

Representation of algebraic cycles by smooth or local complete intersection cycles is well-understood.

New elementary observations on cycle representation problems are provided.

The paper consolidates known results and adds fresh simple insights.

Abstract

In this short note, we simply collect some known results about representing algebraic cycles by various kind of "nice" (e.g. smooth, local complete intersection, products of local complete intersection) algebraic cycles, up to rational equivalence. We also add a few elementary and easy observations on these representation problems that we were not able to locate in the literature.