Double EPW cubes from twisted cubics on Gushel-Mukai fourfolds

TL;DR

This paper explores the geometry of twisted cubics on Gushel-Mukai fourfolds and establishes a connection between double EPW cubes and the Hilbert schemes of these cubics, revealing new geometric structures.

Contribution

It is the first systematic study of twisted cubics on GM fourfolds and links double EPW cubes to the Hilbert scheme of twisted cubics, supporting O'Grady's conjecture.

Findings

Double EPW cube is the MRC quotient of the Hilbert scheme of twisted cubics.

General double EPW cube admits a Lagrangian covering from GM threefolds.

Provides new geometric examples related to hyperkähler manifolds.

Abstract

In this paper, we conduct the first systematic investigation of twisted cubics on Gushel-Mukai (GM) fourfolds. We then study the double EPW cube, a 6-dimensional hyperk\"ahler manifold associated with a general GM fourfold $X$, through the Bridgeland moduli space, and show that it is the maximal rationally connected (MRC) quotient of the Hilbert scheme of twisted cubics on $X$. We also prove that a general double EPW cube admits a covering by Lagrangian subvarieties constructed from the Hilbert schemes of twisted cubics on GM threefolds, which provides a new example for a conjecture of O'Grady.