Schémas de Fano

1996/11/26 by Debarre, O., Manivel, L.
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.alg-geom/9611033

Abstract

Let X be a subvariety of Pⁿ defined by equations of degrees d =(d₁,...,d_s), over an algebraically closed field k of any characteristic. We study properties of the Fano scheme F_r(X) that parametrizes linear subspaces of dimension r contained in X. We prove that F_r(X) is connected and smooth of the expected dimension for n big enough (this was previously known in characteristic 0 or for r=1). Using Bott's theorem, we prove a vanishing theorem for certain bundles on the Grassmannian and use it to calculate the cohomology groups of F_r(X) in degree ≤ dim X-2r, and to prove that F_r(X) is projectively normal in the Grassmannian. Finally, we prove that for n big enough, the rational Chow group A₁(F_r(X)) is of rank 1, and F_r(X) is unirational. All bounds on n are effective.

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