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Intersecting families of vector spaces with maximum covering number

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_q$. Suppose that $\mathscr{F}$ is an intersecting family of $m$-dimensional subspaces of $V$. The covering number of $\mathscr{F}$ is the minimum dimension of a subspace of $V$ which intersects all elements of $\mathscr{F}$. In this paper, we give the tight upper bound for the size of $\mathscr{F}$ whose covering number is $m$, and describe the structure of $\mathscr{F}$ which reaches the upper bound. Moreover, we determine the structure of an maximum intersecting family of singular linear space with the maximum covering number.