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

Carolina Araujo and Cinzia Casagrande

Geometry & Topology 21 (2017) 3009–3045

We describe the geometry of the 2m–dimensional Fano manifold G parametrizing (m1)–planes in a smooth complete intersection Z of two quadric hypersurfaces in the complex projective space 2m+2 for m 1. We show that there are exactly 22m+2 distinct isomorphisms in codimension one between G and the blow-up of 2m at 2m + 3 general points, parametrized by the 22m+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).

Fano varieties, intersection of two quadrics, blow-up of projective spaces, birational geometry
Mathematical Subject Classification 2010
Primary: 14E30, 14J45
Secondary: 14M15, 14N20, 14E05
Received: 14 February 2016
Revised: 29 July 2016
Accepted: 20 November 2016
Published: 15 August 2017
Proposed: Lothar Göttsche
Seconded: Dan Abramovich, Gang Tian
Carolina Araujo
Instituto de Matemática Pura e Aplicada
22460-320 Rio de Janeiro
Cinzia Casagrande
Dipartimento di Matematica
Università di Torino
I-10123 Torino