Intersections of two Grassmannians in $\mathbf{P}^9$
Abstract
We study the intersection of two copies of $\mathrm{Gr}(2,5)$ embedded in $\mathbf{P}^9$, and the intersection of the two projectively dual Grassmannians in the dual projective space. These intersections are deformation equivalent, derived equivalent CalabiYau threefolds. We prove that generically they are not birational. As a consequence, we obtain a counterexample to the birational Torelli problem for CalabiYau threefolds. We also show that these threefolds give a new pair of varieties whose classes in the Grothendieck ring of varieties are not equal, but whose difference is annihilated by a power of the class of the affine line. Our proof of nonbirationality involves a detailed study of the moduli stack of CalabiYau threefolds of the above type, which may be of independent interest.
 Publication:

arXiv eprints
 Pub Date:
 July 2017
 DOI:
 10.48550/arXiv.1707.00534
 arXiv:
 arXiv:1707.00534
 Bibcode:
 2017arXiv170700534B
 Keywords:

 Mathematics  Algebraic Geometry
 EPrint:
 30 pages, minor changes