We describe the geometry of the –dimensional Fano manifold parametrizing –planes in a smooth complete intersection of two quadric hypersurfaces in the complex projective space for . We show that there are exactly distinct isomorphisms in codimension one between and the blow-up of at general points, parametrized by the distinct –planes contained in , and describe these rational maps explicitly. We also describe the cones of nef, movable and effective divisors of , as well as their dual cones of curves. Finally, we determine the automorphism group of .
These results generalize to arbitrary even dimension the classical description of quartic del Pezzo surfaces ().
"On the Fano variety of linear spaces contained in two odd-dimensional quadrics." Geom. Topol. 21 (5) 3009 - 3045, 2017. https://doi.org/10.2140/gt.2017.21.3009