Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Benson Farb

TL;DR

The paper proves that the intermediate Jacobian $J(X)$ of the Klein quartic 3-fold $X$ is not a product of Jacobians of curves, hence by the Clemens-Griffiths criterion $X$ is irrational, and the general smooth quartic 3-fold is also irrational. The argument exploits the explicit symmetry group $G=\mathbb{Z}/61\mathbb{Z}\rtimes \mathbb{Z}/5\mathbb{Z}$ acting on $X$ and its p.p.a.v. $J(X)$, combined with Beauville-type reasoning about decompositions of $J(X)$ into irreducible factors. A key step extracts a factor $B\cong {\rm Jac}(C_1)$ on which the $61$-torsion acts faithfully, then uses Riemann bounds to force ${\rm genus}(C_1)\ge 15$, and a group-theoretic contradiction (metacyclic bound) shows no such decomposition can occur. This yields a shorter, symmetry-driven irrationality proof for quartic 3-folds, complementing classical approaches such as those of Iskovskih-Manin.

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.

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

TL;DR

The paper proves that the intermediate Jacobian of the Klein quartic 3-fold is not a product of Jacobians of curves, hence by the Clemens-Griffiths criterion is irrational, and the general smooth quartic 3-fold is also irrational. The argument exploits the explicit symmetry group acting on and its p.p.a.v. , combined with Beauville-type reasoning about decompositions of into irreducible factors. A key step extracts a factor on which the -torsion acts faithfully, then uses Riemann bounds to force , and a group-theoretic contradiction (metacyclic bound) shows no such decomposition can occur. This yields a shorter, symmetry-driven irrationality proof for quartic 3-folds, complementing classical approaches such as those of Iskovskih-Manin.

Abstract

We prove that the intermediate Jacobian of the Klein quartic -fold 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 , as well as the general smooth quartic -fold, is irrational. These corollaries were known: Iskovskih-Manin \cite{IM} proved that every smooth quartic -fold is irrational. However, the method of proof here is different than that of \cite{IM} and is significantly simpler.
Paper Structure (2 sections, 4 theorems, 12 equations)

This paper contains 2 sections, 4 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['Theorem:IJ1']}

Key Result

Theorem 1.1

The intermediate Jacobian $\mathop{\mathrm{J}}\nolimits(X)$ of the Klein quartic $3$-fold $X$ is not isomorphic, as a principally polarized abelian variety, to a product of Jacobians of smooth curves.

Theorems & Definitions (7)

  • Theorem 1.1: Intermediate Jacobian
  • Corollary 1.2: Irrationality of Klein
  • Corollary 1.3: Irrationality is general
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['Theorem:IJ1']}
  • Remark 2.2