Deformation Classification of Real Non-singular Cubic Threefolds with a Marked Line

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.