Covering by complements of subspaces, II

Let $V$ be an $n$-dimensional vector space over an algebraically closed field $K$. Define $\gamma (k,n,K)$ to be the least positive integer $t$ for which there exists a family $E_{1}, E_{2}, \dots , E_{t}$ of $k$-dimensional subspaces of $V$ such that every $(n-k)$-dimensional subspace $F$ of $V$ has at least one complement among the $E_{i}$'s. Using algebraic geometry we prove that $\gamma (k,n,K) = k(n-k) +1$.