Matrix factorizations and intermediate Jacobians of cubic threefolds

TL;DR

This paper links the intermediate Jacobian of smooth cubic threefolds to moduli spaces of sheaves and matrix factorizations, enabling generalizations to singular and reducible cases, and studying degenerations.

Contribution

It introduces a new description of the moduli space of sheaves via matrix factorizations, extending the framework to singular and reducible cubic threefolds.

Findings

Identification of the moduli space with matrix factorizations.

Extension of the framework to singular and reducible cubic threefolds.

Analysis of degenerations of the moduli space in reducible cases.

Abstract

Results due to Druel and Beauville show that the blowup of the intermediate Jacobian of a smooth cubic threefold X in the Fano surface of lines can be identified with a moduli space of semistable sheaves of Chern classes c_1=0, c_2=2, c_3=0 on X. Here we further identify this space with a space of matrix factorizations. This has the advantage that this description naturally generalizes to singular and even reducible cubic threefolds. In this way, given a degeneration of X to a reducible cubic threefold X_0, we obtain an associated degeneration of the above moduli spaces of semistable sheaves.