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

Abstract

In this paper we describe the geometry of the 2m-dimensional Fano manifold G parametrizing (m − 1)-planes in a smooth complete intersection Z of two quadric hypersurfaces in the complex projective space P 2m+2 , for m ≥ 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).

On Fano Schemes of Complete Intersections

“…In conclusion we find the formula (38) deg(Σ c (d, r)) = − I={i,j,k}∈I 3 {a,b}∈I (2) η(t i , t j , t k ) (t i t j t k ) r−2 {α,β}∈I (2) \{a,b} (t α + t β )…”

“…Then the Fano scheme F k (X) is reduced and finite of cardinality 2 2k+2 ([42, Ch. 2]), whereas F k−1 (X) is a rational Fano variety of dimension 2k and index 1, whose Picard number is ρ = 2k + 4, see [2], [13], and the references therein.…”

“…The characters of the T -action on E 2,Γ have weights t α + t β with {α, β} ∈ I (2) \ {a, b}, where x a x b = 0 is the equation of Γ in Π. Then e Γ = (−1) 3(r−2) (t i t j t k ) r−2 {α,β}∈I (2) \{a,b} (t α + t β ) .…”

A note on the fibres of Mori fibre spaces

“…We now focus on the first example. We use the results of [AC17] and [Rei72, Section 3]. Take an even integer n = 2m ≥ 2 and consider a smooth complete intersection Z of two quadrics in P n+2 .…”

“…Let F m−1 = F m−1 (Z) be the variety of (m − 1)-planes in Z. This is a smooth Fano variety of dimension n. It can be seen as a higher dimensional generalisation of the quartic del Pezzo surface and has been extensively studied in the recent work [AC17].…”