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

Let $\text{G}(k,r)$ be the Grassmannian of $k$–subspaces in ${\text{Proj}}^r$ embedded in ${\text{Proj}}^{N(k,r)}$, with $N(k,r)=\left(\genfrac{}{}{0px}{}{r+1}{k+1}\right)-1$, via the Plücker embedding. In this paper, extending some classical results by Gallarati (see \cite {Gal1,Gal2}), we give a sharp upper bound for the number of independent sections of $H^0(\text{G}(k,r),{\text{Ø}}_{\text{G}(k,r)}(m))$ vanishing on a subvariety $X$ of $\text{G}(k,r)$ such that the union of the $k$–subspaces corresponding to the points of $X$ spans ${\text{Proj}}^r$.