Moduli spaces of quadratic complexes and their singular surfaces

Abstract

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