Geometry & Topology

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

Carolina Araujo and Cinzia Casagrande

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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).

Article information

Geom. Topol., Volume 21, Number 5 (2017), 3009-3045.

Received: 14 February 2016
Revised: 29 July 2016
Accepted: 20 November 2016
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14E30: Minimal model program (Mori theory, extremal rays) 14J45: Fano varieties
Secondary: 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 14N20: Configurations and arrangements of linear subspaces 14E05: Rational and birational maps

Fano varieties intersection of two quadrics blow-up of projective spaces birational geometry


Araujo, Carolina; Casagrande, Cinzia. On the Fano variety of linear spaces contained in two odd-dimensional quadrics. Geom. Topol. 21 (2017), no. 5, 3009--3045. doi:10.2140/gt.2017.21.3009.

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