Deformation classification of real non-singular cubic threefolds with a marked line

TL;DR

This paper classifies the deformation types of pairs consisting of a real non-singular cubic threefold and a marked real line, revealing 18 connected components with explicit geometric and topological interpretations.

Contribution

It provides a complete classification of the deformation components of real cubic threefolds with a marked line, linking them to monodromy orbits and deformation classes of plane quintic curves.

Findings

18 connected components of the space of pairs (X, l)

Explicit interpretations relate components to monodromy orbits and deformation classes

Characterization of components via deformation classes of plane quintic curves

Abstract

We prove that the space of pairs $(X,l)$ formed by a real non-singular cubic hypersurface $X\subset P^4$ with a real line $l\subset X$ has 18 connected components and give for them several quite explicit interpretations. The first one relates these components to the orbits of the monodromy action on the set of connected components of the Fano surface $F_\mathbb{R}(X)$ formed by real lines on $X$. For another interpretation we associate with each of the 18 components a well defined real deformation class of real non-singular plane quintic curves and show that this deformation class together with the real deformation class of $X$ characterizes completely the component.