Stable maps of curves and algebraic equivalence of 1-cycles

TL;DR

This paper demonstrates that algebraic equivalence of stable map images lifts to deformation equivalence, providing new insights into the structure of 1-cycle groups on rationally connected varieties and their behavior under field extensions.

Contribution

It establishes a lifting property for algebraic equivalence to deformation equivalence of stable maps and analyzes the structure of 1-cycle groups over various fields.

Findings

Kernel of $A_1(X_k) o A_1(X_K)$ is at most ${f Z}/2{f Z}$.

Image of $A_1(X) o A_1(ar{X})$ equals Galois invariants when $k$ is finite.

Provides new tools for understanding 1-cycles on rationally connected varieties.

Abstract

We show that algebraic equivalence of images of stable maps of curves lifts to deformation equivalence of the stable maps. The main applications concern $A_1(X)$, the group of 1-cycles modulo algebraic equivalence, for smooth, separably rationally connected varieties. If $K/k$ is an algebraic extension, then the kernel of $A_1(X_k)\to A_1(X_K)$ is at most ${\mathbb Z}/2{\mathbb Z}$. If $k$ is finite, then the image equals the subgroup of Galois invariant cycles. This paper replaces Sections~2--3 of 2211.15915.v.1 and Sections~2--3 of 2211.15911.v.1. The other Sections are retained in the revised versions of these papers.