The Fermat cubic and monodromy of lines

TL;DR

This paper investigates the geometry of lines on cubic threefolds and fourfolds, revealing the structure of second type lines and monodromy groups, with implications for algebraic geometry and Fano schemes.

Contribution

It provides an explicit description of the second type lines on Fermat cubic fourfolds and computes the monodromy groups for general cubic threefolds, advancing understanding of their geometric properties.

Findings

Voisin map is birational over the second type locus

Monodromy groups are the full symmetric group for general cubic threefolds

Explicit description of the second type lines on Fermat cubic fourfolds

Abstract

In this paper we study properties of the locus of second type lines of a general cubic threefold and fourfold. By analysing the geometry of the Fano scheme of lines of the Fermat cubic fourfold and in particular giving an explicit description of the locus of second type lines, we deduce that the Voisin map is birational over the second type locus. For a general cubic threefold, by studying properties of the second type locus again, we compute that various natural geometric monodromy groups are the full symmetric group.