The generalized Plücker–Klein map

The intersection of two quadrics is called a biquadric. If we mark a non-singular quadric in the pencil of quadrics through a given biquadric, then the given biquadric is called a marked biquadric. In the classical papers of Plücker and Klein, a Kummer surface was canonically associated with every three-dimensional marked biquadric (that is, with a quadratic line complex provided that the Plücker–Klein quadric is marked). In Reid's thesis, this correspondence was generalized to odd-dimensional marked biquadrics of arbitrary dimension $\geqslant 3$. In this case, a Kummer variety of dimension $g$ corresponds to every biquadric of dimension $2g-1$. Reid only constructed the generalized Plücker–Klein correspondence. This map was not studied later. The present paper is devoted to a partial solution of the problem of creating the corresponding theory.