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