Irrationality of the general smooth quartic $3$-fold using intermediate Jacobians

TL;DR

This paper demonstrates that the general smooth quartic threefold is irrational by analyzing its intermediate Jacobian, providing a simpler proof than previous methods and confirming its non-decomposability into Jacobians of curves.

Contribution

The paper introduces a new, simpler proof of the irrationality of the general smooth quartic threefold using intermediate Jacobians.

Findings

The intermediate Jacobian of the Klein quartic threefold is not isomorphic to a product of Jacobians.

The general smooth quartic threefold is irrational.

The proof offers a simpler alternative to previous methods.

Abstract

We prove that the intermediate Jacobian of the Klein quartic $3$-fold $X$ is not isomorphic, as a principally polarized abelian variety, to a product of Jacobians of curves. As corollaries we deduce (using a criterion of Clemens-Griffiths) that $X$, as well as the general smooth quartic $3$-fold, is irrational. These corollaries were known: Iskovskih-Manin \cite{IM} proved that every smooth quartic $3$-fold is irrational. However, the method of proof here is different than that of \cite{IM} and is significantly simpler.