Moduli spaces of quadratic complexes and their singular surfaces

We construct the coarse moduli space ${\text{cM}}_{qc}(\sigma )$ of quadratic line complexes with a fixed Segre symbol $\sigma$ as well as the moduli space ${\text{cM}}_{ss}(\sigma )$ of the corresponding singular surfaces. We show that the map associating to a quadratic line complex its singular surface induces a morphism $\pi :{\text{cM}}_{qc}(\sigma )\text{ra}{\text{cM}}_{ss}(\sigma )$. Finally we deduce that the varieties of cosingular quadratic line complexes are almost always curves.