Schémas de Fano

Details Schémas de Fano Schémas de Fano

1996-11-26

arxiv.org/abs/alg-geom/9611033v1

Mathematics

Algebraic Geometry

PlainTeX v 1.2, 22 pages, in French. Run it twice to get cross-references right

Scientific paper

Let X be a subvariety of $P^n$ defined by equations of degrees $ d =(d_1,...,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 $\le \dim X-2r$, 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 $A_1(F_r(X))$ is of rank 1, and $F_r(X)$ is unirational. All bounds on n are effective.

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