  We use the density matrix renormalization group method to study the ground
state properties of an antiferromagnetic spin-$1$ chain with a next-nearest
neighbor exchange $J_2 ~$ and an alternation $\delta$ of the nearest neighbor
exchanges. We find a line running from a gapless point at $(J_2 , \delta) = (0,
0.25 \pm 0.01)$ upto an almost gapless point at $(0.725 \pm 0.01, 0$ such that
the open chain ground state is $4$-fold degenerate below the line and is unique
above it. A disorder line $2 J_2 + \delta = 1$ runs from $\delta =0$ to about
$\delta =0.136$. To the left of this line, the peak in the structure factor
$S(q)$ is at $\pi$, while to the right of the line, it is at less than $\pi$.
