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

Araujo, Carolina; Casagrande, Cinzia

doi:10.2140/gt.2017.21.3009

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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

DOI: 10.2140/gt.2017.21.3009

Abstract

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

Keywords

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

Mathematical Subject Classification 2010

Primary: 14E30, 14J45

Secondary: 14M15, 14N20, 14E05

References

Publication

Received: 14 February 2016

Revised: 29 July 2016

Accepted: 20 November 2016

Published: 15 August 2017

Proposed: Lothar Göttsche

Seconded: Dan Abramovich, Gang Tian

Authors

Carolina Araujo
Instituto de Matemática Pura e Aplicada 22460-320 Rio de Janeiro Brazil

Cinzia Casagrande
Dipartimento di Matematica Università di Torino I-10123 Torino Italy

Volume 21, issue 5 (2017)