Moduli spaces of quadratic complexes and their singular surfaces

We construct the coarse moduli space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{M}}_{qc}(\sigma)$$\end{document} of quadratic line complexes with a fixed Segre symbol σ as well as the moduli space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{M}}_{ss}(\sigma)$$\end{document} of the corresponding singular surfaces. We show that the map associating to a quadratic line complex its singular surface induces a morphism \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi: {\mathcal{M}}_{qc}(\sigma) \rightarrow {\mathcal{M}}_{ss}(\sigma)$$\end{document} . Finally we deduce that the varieties of cosingular quadratic line complexes are almost always curves.