On the Fano variety of linear spaces contained in two odd-dimensional quadrics

摘要

We describe the geometry of the 2m-dimensional Fano manifold G parametrizing (m-1)-planes in a smooth complete intersection Z of two quadric hypersurfaces in the complex projective space P2m+2 for m >= 1. We show that there are exactly 2(2m+2) distinct isomorphisms in codimension one between G and the blow-up of P-2m at 2m + 3 general points, parametrized by the 2(2m+2) distinct m-planes contained in Z, and describe these rational maps explicitly. We also describe the cones of nef, movable and effective divisors of G, as well as their dual cones of curves. Finally, we determine the automorphism group of G.