  We consider two varieties associated to a web of quadrics W in the projective
space of dimension 7. One is the base locus and the second one is the double
cover of the three dimensional projective space branched along the determinant
surface of W. We show that small resolutions of these varieties are Calabi-Yau
manifolds. We compute their Betti numbers and show that they are not birational
in the generic case. The main result states that if the base locus of W
contains a plane then in the generic case the two varieties are birational.
