#### Volume 21, issue 5 (2017)

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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
##### Abstract

We describe the geometry of the $2m$–dimensional Fano manifold $G$ parametrizing $\left(m-1\right)$–planes in a smooth complete intersection $Z$ of two quadric hypersurfaces in the complex projective space ${ℙ}^{2m+2}$ for $m\ge 1$. 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 ($m=1$).

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##### 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