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

I will 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 of dimension 2m+2. We will see 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. This birational maps allow to determine the cones of nef, movable and effective divisors of G, and the automorphism group of G. This generalizes to arbitrary even dimension the classical description of quartic del Pezzo surfaces (m = 1). This is a joint work with Carolina Araujo (IMPA).