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

TL;DR

This paper explores the geometry of a Fano manifold associated with (m-1)-planes in a complete intersection of two quadrics, revealing explicit isomorphisms, divisor cones, and automorphisms, generalizing classical results for quartic del Pezzo surfaces.

Contribution

It provides a detailed description of the Fano manifold's geometry, including isomorphisms, divisor cones, and automorphisms, extending classical results to higher even dimensions.

Findings

Exactly 2^{2m+2} isomorphisms in codimension one between G and a blow-up of projective space.

Explicit descriptions of nef, movable, and effective divisor cones.

Determination of the automorphism group of G.

Abstract

In this paper 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 P^{2m+2}, for m>0. 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. These results generalize to arbitrary even dimension the classical description of quartic del Pezzo surfaces (m=1).