On the maximum number of complexes of a given degree containing subvarieties of Grassmannians

Let \({\mathbb G}(k,r)\) be the Grassmannian of k-subspaces in \({\mathbb P}^r\) embedded in \({\mathbb P}^{N(k,r)}\), with \(N(k,r)={{r+1}\atopwithdelims (){k+1}}-1\), via the Plücker embedding. In this paper, extending some classical results by Gallarati (see Gallarati in Rend Accad Naz Lincei Ser VIII 14(2):213–220, 1953, Rend Accad Naz Lincei Ser VIII 14(3):408–412, 1953), we give a sharp upper bound for the number of independent sections of \(H^0({\mathbb G}(k,r), {\mathcal O}_{{\mathbb G}(k,r)}(m))\) vanishing on a subvariety X of \({\mathbb G}(k,r)\) such that the union of the k-subspaces corresponding to the points of X spans \({\mathbb P}^r\).