Double EPW cubes from twisted cubics on Gushel-Mukai fourfolds
Soheyla Feyzbakhsh, Hanfei Guo, Zhiyu Liu, Shizhuo Zhang
TL;DR
This work establishes that, for a general Gushel–Mukai fourfold $X$, the associated hyperkähler sixfold $ ilde C_X$ (the double EPW cube) is the maximal rationally connected quotient of the Hilbert scheme $ ext{Hilb}_X^{3t+1}$ of twisted cubics on $X$. The approach uses Bridgeland moduli spaces of stable objects in the Kuznetsov component $ ext{Ku}(X)$ to connect $ ilde C_X$ with moduli spaces $M^X_{oldsymbol{ au}_X}(1,-1)$ and to the period data, showing birational equivalences that realize $ ilde C_X$ as an MRC quotient. The paper also constructs a Lagrangian covering family on $ ilde C_X$ by exploiting Hilbert schemes of twisted cubics on GM threefolds and the Bridgeland-moduli framework, providing a new instance supporting O'Grady's conjecture. These results unify categorical stability with classical hyperkähler geometry, enriching the landscape of explicit hyperkähler manifolds arising from GM geometries and their moduli spaces.
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 hyperkähler 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.