Lines, Twisted Cubics on Cubic Fourfolds, and the Monodromy of the Voisin Map
Franco Giovenzana, Luca Giovenzana
TL;DR
The paper proves that, for a general cubic fourfold $Y$, the monodromy group of Voisin's degree-$16$ self-map $\psi$ on the Fano variety $F$ of lines is the full symmetric group on $16$ elements. The authors develop a framework linking $\psi$ to Voisin's degree-$6$ map $\varphi:F\times F\dashrightarrow Z$ and to the LLSvS hyperk"ahler variety $Z$ with its antisymplectic involution, using the fixed locus $W$ and a dominating subvariety $P\subset F\times F$ to control the monodromy via ramification and transitivity arguments. They analyze nodal quintic discriminants, Cayley cubics and their degenerations, and simple elliptic cubic surfaces to understand the geometry of twisted cubics and generalised twisted cubics, which feeds into the monodromy computation. The main result follows from showing $P$ is irreducible and that the monodromy contains a transposition and is $2$-transitive, yielding maximal monodromy $\mathrm{Mon}_{\widehat{\psi}}=S_{16}$. This work reveals deep connections between line geometry on cubic fourfolds, the LLSvS moduli space, and monodromy phenomena for finite maps on hyperk"ahler manifolds, with implications for the global behavior of non-birational self-maps in hyperk"ahler geometry.
Abstract
For a general cubic fourfold $Y$ with associated Fano variety of lines $ F $, we show that the monodromy group of the finite degree 16 rational Voisin self-map $ψ\colon F \dashrightarrow F$ is maximal. To achieve this, we investigate the intriguing interplay between $ ψ$ and the fixed locus of the antisymplectic involution on the LLSvS variety $ Z $, examined via the degree 6 Voisin map $F \times F \dashrightarrow Z $.