Lines, Twisted Cubics on Cubic Fourfolds, and the Monodromy of the Voisin Map

TL;DR

This paper proves that the monodromy group of a specific rational self-map on the Fano variety of lines of a general cubic fourfold is maximal, using the interplay between this map and the fixed locus of an antisymplectic involution.

Contribution

It establishes the maximality of the monodromy group for the Voisin map on the Fano variety of lines of a cubic fourfold, connecting it with the geometry of the LLSvS variety.

Findings

Monodromy group of the Voisin map is maximal.

Relationship between the Voisin map and the antisymplectic involution.

Analysis of the fixed locus on the LLSvS variety.

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 $\psi \colon F \dashrightarrow F$ is maximal. To achieve this, we investigate the intriguing interplay between $ \psi $ 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 $.