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We use the density matrix renormalization group method to study the groundstate `phase' diagram and some low-energy properties of isotropicantiferromagnetic spin-$1 \over 2$ and spin-$1$ chains with a next-nearestneighbor exchange $J_2 ~$ and an alternation $\delta$ of the nearest neighborexchanges. In the spin-$1 \over 2$ chain, the system is gapless for $\delta=0$and $J_2 < J_{2c} =0.241$, and is gapped everywhere else in the $J_2 - \delta$plane. At $J_{2c}$, for small $\delta$, the gap increases as $\delta^{\alpha}$,where $\alpha = 0.667 \pm 0.001$. $2J_2 + \delta = 1$ is a disorder line. Tothe left of this line, the structure factor $S(q)$ peaks at $q_{max} = \pi$(Neel `phase'), while to the right, $q_{max}$ decreases from $\pi$ to $\pi/2$(spiral `phase') as $J_2$ increases. There is also a `$\uparrow \uparrow\downarrow \downarrow$ phase' for large values of both $J_2$ and $\delta$. Inthe spin-$1$ case, we find a line running from a gapless point at $(J_2 ,\delta) = (0,0.25 \pm 0.01)$ upto a `gapless' point at $(0.73 \pm 0.005,0)$such that the open chain ground state is four-fold degenerate below the lineand is unique above it. There is a disorder line in this case also and it hasthe same equation as in the spin-$1 \over 2$ case, but the line ends at about$\delta =0.136$. Similar to the spin-$1 \over 2$ case, to the left of thisline, the peak in the structure factor is at $\pi$ (Neel `phase'), while to theright of the line, it is at less than $\pi$ (spiral `phase'). For $\delta =1$,the system corresponds to a spin ladder and the system is gapped for all valuesof the interchain coupling for both spin-$1 \over 2$ and spin-$1$ ladders.Comment: 16 pages in latex; 9 figures available on reques

A comparative study of the phase diagrams of spin- and spin-1 antiferromagnetic chains with dimerization and frustration