Minimal classes on the intermediate Jacobian of a generic cubic threefold

TL;DR

This paper proves that for a generic smooth cubic threefold, the Fano surface is uniquely characterized as the only surface of minimal cohomology class within its associated intermediate Jacobian.

Contribution

It establishes the uniqueness of the Fano surface as the sole minimal class surface in the intermediate Jacobian of a generic cubic threefold.

Findings

Fano surface embeds in the intermediate Jacobian with minimal class.

Uniqueness of the Fano surface as the only minimal class surface for generic cubic threefolds.

Supports Clemens-Griffiths theorem with a uniqueness result.

Abstract

Let X be a smooth cubic threefold. We can associate two objects to X: the intermediate Jacobian J and the Fano surface F parametrising lines on X. By a theorem of Clemens and Griffiths, the Fano surface can be embedded in the intermediate Jacobian and the cohomology class of its image is minimal. In this paper we show that if X is generic, the Fano surface is the only surface of minimal class in J.