Generalized twisted cubics on a cubic fourfold as a moduli space of stable objects

TL;DR

This paper explores the moduli space of stable objects associated with twisted cubic curves on a cubic fourfold, revealing new geometric relationships and stability conditions.

Contribution

It establishes isomorphisms between various moduli spaces of stable sheaves and objects, connecting geometric and derived category perspectives.

Findings

The blow-up along the cubic is isomorphic to a moduli space component.

The fourfold is a component of a moduli space of tilt-stable and Bridgeland stable objects.

Wall-crossing describes the contraction between the moduli spaces.

Abstract

We revisit the work of Lehn-Lehn-Sorger-van Straten on twisted cubic curves in a cubic fourfold not containing a plane in terms of moduli spaces. We show that the blow-up $Z'$ along the cubic of the irreducible holomorphic symplectic eightfold $Z$, described by the four authors, is isomorphic to an irreducible component of a moduli space of Gieseker stable torsion sheaves or rank three torsion free sheaves. For a very general such cubic fourfold, we show that $Z$ is isomorphic to a connected component of a moduli space of tilt-stable objects in the derived category and to a moduli space of Bridgeland stable objects in the Kuznetsov component. Moreover, the contraction between $Z'$ and $Z$ is realized as a wall-crossing in tilt-stability. Finally, $Z$ is birational to an irreducible component of Gieseker stable aCM bundles of rank six.