GV-subschemes and their embeddings in principally polarized abelian varieties

TL;DR

This paper proves that GV-subschemes embedded in principally polarized abelian varieties do not factor through nontrivial isogenies, and applies this to give a new proof of a classical result relating cubic threefolds and their Fano surfaces.

Contribution

It establishes a new property of GV-subschemes regarding their embeddings and provides an alternative proof of a key theorem connecting cubic threefolds and their intermediate Jacobians.

Findings

GV-subschemes do not factor through nontrivial isogenies

New proof of Clemens--Griffiths theorem on cubic threefolds

Identification of intermediate Jacobian with Albanese of Fano surface

Abstract

We prove that the embedding of a $GV$-subscheme in a principally polarized abelian variety does not factor through any nontrivial isogeny. As an application, we present a new proof of a theorem of Clemens--Griffiths identifying the intermediate Jacobian of a smooth cubic threefold in $\mathbf{P}^4$ to the Albanese variety of its Fano surface of lines.