# M-regularity of the Fano surface

## TL;DR

This paper proves that the structure sheaf of the Fano surface of lines on a smooth cubic threefold is M-regular, supporting a conjecture linking minimal cohomology class subvarieties to M-regularity in abelian varieties.

## Contribution

It demonstrates that the Fano surface's structure sheaf satisfies M-regularity, providing evidence for the conjecture relating minimal cohomology class and M-regularity.

## Key findings

- Fano surface of lines on a cubic threefold has M-regular structure sheaf.
- Supports the conjecture linking minimal cohomology class to M-regularity.
- Enhances understanding of subvarieties in abelian varieties.

## Abstract

Let $(A,\Theta)$ be a principally polarised abelian variety, and let Y be a subvariety. Pareschi and Popa conjectured that Y has minimal cohomology class if and only if the structure sheaf of Y satisfies a property that they call M-regularity. Let now X be a smooth cubic threefold. By a classical result due to Clemens and Griffiths, its intermediate Jacobian J(X) is a principally polarised abelian variety; furthermore the Fano surface of lines on X can be embedded in J(X) and has minimal cohomology class. In this short note we show that its structure sheaf is M-regular.