Birational maps between Calabi-Yau manifolds associated to webs of quadrics

TL;DR

This paper studies two Calabi-Yau manifolds derived from a web of quadrics in projective space, analyzing their birational relationships and topological properties.

Contribution

It establishes conditions under which these two Calabi-Yau varieties are birational, especially when the base locus contains a plane, and computes their Betti numbers.

Findings

Small resolutions yield Calabi-Yau manifolds.

The two varieties are not birational in general.

They become birational if the base locus contains a plane.

Abstract

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.