Geometric representability of 1-cycles on rationally connected threefolds

TL;DR

This paper proves that for any rationally connected threefold, there exists a smooth surface and a family of 1-cycles that induce an Abel-Jacobi isomorphism, extending known results beyond specific Fano threefolds.

Contribution

It establishes the geometric representability of the intermediate Jacobian for all rationally connected threefolds, generalizing previous results for special classes.

Findings

Existence of a smooth surface and 1-cycle family inducing Abel-Jacobi isomorphism.

Extension of known cases from Fano threefolds to all rationally connected threefolds.

Provides a new geometric framework for understanding intermediate Jacobians.

Abstract

We prove that for any rationally connected threefold $X$, there exists a smooth projective surface $S$ and a family of $1$-cycles on $X$ parameterized by $S$, inducing an Abel-Jacobi isomorphism ${\rm Alb}(S)\cong J^3(X)$. This statement was previously known for some classes of smooth Fano threefolds.