  In this paper, we study the curve cone of an almost complex $4$-manifold
which is tamed by a symplectic form. In particular, we prove the cone theorem
as in Mori theory for all such manifolds using the Seiberg-Witten theory. For
small rational surfaces and minimal ruled surfaces, we study the configuration
of negative curves. We define abstract configuration of negative curves, which
records the homological and intersection information of curves. Combinatorial
blowdown is the main tool to study these configurations. As an application of
our investigation of the curve cone, we prove the Nakai-Moishezon type duality
for all almost K\"ahler structures on $\mathbb CP^2\#k\overline{\mathbb CP^2}$
with $k\le 9$ and minimal ruled surfaces with a negative curve. This is proved
using a version of Gram-Schmidt orthogonalization process for the $J$-tamed
symplectic inflation.
