Sch\'emas de Fano

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.