Cremona transformations associated to quadratic complexes

Abstract

To a general quadratic line complex and any pair of different lines in it one can associate a cubo-cubic Cremona transformation of projective 3-space in a natural way. This gives a map $$\gamma $$ from the space of pairs of lines in general quadratic complexes to the space of equivalence classes of these cubo-cubic transformations. We compute the dimension of a general fibre of $$\gamma $$ as well as the codimension of subspace of the transformations arizing in this way in the corresponding space of Cremona transformations.