Fano schemes of linear spaces on hypersurfaces (Q1360911)

Let \(X\) be a generic hypersurface of degree \(d\) in \(\mathbb{P}^n\) and let \(F\) be the Fano scheme of the \(k\)-planes contained in \(X\). The author proves that \(F\) is smooth of dimension \[ \varphi(n,k,d) =(k+1) (n-k)- {d+k \choose k} \] and moreover \(F\) is connected if \(\varphi (n,k,d) >0\). The proof is essentially based on the study of the critical points of the projection \(p:I\to H^0 (\mathbb{P}^n, O(d))\), where \(I=\{(\pi,X) \mid\pi \subset X\}\) is the incidence correspondence and the result follows from an accurate analysis of the dimension of some suitable determinantal varieties.