The structure of the minimum size supertail of a subspace partition

Let $V=V(n,q)$ denote the vector space of dimension $n$ over the finite field with $q$ elements. A subspace partition ${\mathcal P}$ of $V$ is a collection of nontrivial subspaces of $V$ such that each nonzero vector of $V$ is in exactly one subspace of ${\mathcal P}$. For any integer $d$, the $d$-supertail of ${\mathcal P}$ is the set of subspaces in ${\mathcal P}$ of dimension less than $d$, and it is denoted by $ST$. Let $\sigma_q(n,t)$ denote the minimum number of subspaces in any subspace partition of $V$ in which the largest subspace has dimension $t$. It was shown by Heden et al. that $|ST|\geq \sigma_q(d,t)$, where $t$ is the largest dimension of a subspace in $ST$. In this paper, we show that if $|ST|=\sigma_q(d,t)$, then the union of all the subspaces in $ST$ constitutes a subspace under certain conditions.