Geometry & Topology
- Geom. Topol.
- Volume 21, Number 5 (2017), 3009-3045.
On the Fano variety of linear spaces contained in two odd-dimensional quadrics
We describe the geometry of the –dimensional Fano manifold parametrizing –planes in a smooth complete intersection of two quadric hypersurfaces in the complex projective space for . We show that there are exactly distinct isomorphisms in codimension one between and the blow-up of at general points, parametrized by the distinct –planes contained in , and describe these rational maps explicitly. We also describe the cones of nef, movable and effective divisors of , as well as their dual cones of curves. Finally, we determine the automorphism group of .
These results generalize to arbitrary even dimension the classical description of quartic del Pezzo surfaces ().
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
Permanent link to this document
Digital Object Identifier
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
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. https://projecteuclid.org/euclid.gt/1510859280