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

Abstract

We describe the geometry of the $2m$–dimensional Fano manifold $G$ parametrizing $\left(m-1\right)$–planes in a smooth complete intersection $Z$ of two quadric hypersurfaces in the complex projective space ${\mathbb{P}}^{2m+2}$ for $m\ge 1$. We show that there are exactly $2^{2m+2}$ distinct isomorphisms in codimension one between $G$ and the blow-up of ${\mathbb{P}}^{2m}$ 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 $G$.

These results generalize to arbitrary even dimension the classical description of quartic del Pezzo surfaces ($m=1$).