  We inspect the BNSR-invariants $\Sigma^m(P_n)$ of the pure braid groups
$P_n$, using Morse theory. The BNS-invariants $\Sigma^1(P_n)$ were previously
computed by Koban, McCammond and Meier. We prove that for any $3\le m\le n$,
the inclusion $\Sigma^{m-2}(P_n)\subseteq \Sigma^{m-3}(P_n)$ is proper, but
$\Sigma^\infty(P_n)=\Sigma^{n-2}(P_n)$. We write down explicit character
classes in each relevant $\Sigma^{m-3}(P_n)\setminus \Sigma^{m-2}(P_n)$. In
particular we get examples of normal subgroups $N\le P_n$ with
$P_n/N\cong\mathbb{Z}$ such that $N$ is of type $F_{m-3}$ but not $F_{m-2}$,
for all $3\le m\le n$.
