Lines, Twisted Cubics on Cubic Fourfolds, and the Monodromy of the Voisin Map (2412.07483v1)

Abstract: For a cubic fourfold $ Y $ with associated Fano variety of lines $ F $, we establish several properties of the finite-degree 16 self-rational map $ \psi \colon F \dashrightarrow F $ introduced by Voisin. We begin by analyzing the singularities of the nodal quintic with 16 nodes associated with a general line under the specialization to a line in the branch locus of $ \psi $. This approach reveals that the ramification of the natural resolution of indeterminacy of $ \psi $ is simple. The main part of the paper focuses on 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 $. As an application, we show that the monodromy of $ \psi $ is maximal.