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

Araujo, Carolina¹ ; Casagrande, Cinzia²

¹ Instituto de Matemática Pura e Aplicada, 22460-320 Rio de Janeiro, Brazil
² Dipartimento di Matematica, Università di Torino, I-10123 Torino, Italy

Geometry & topology, Tome 21 (2017) no. 5, pp. 3009-3045.

Résumé

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 ${\mathbb{P}}^{2m+2}$ 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 ${\mathbb{P}}^{2m}$ 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 ($m=1$).

Détail

DOI : 10.2140/gt.2017.21.3009

Classification : 14E30, 14J45, 14M15, 14N20, 14E05
Keywords: Fano varieties, intersection of two quadrics, blow-up of projective spaces, birational geometry

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