  We study the axio-dilaton and complex structure stabilization for a generic
one-parameter Calabi-Yau (CY) compactification. We focus on near the conifold
regions and consider a strongly warped metric. We analyze numerically the case
of the mirror of the quintic CY. The axio-dilaton $\tau$ and complex structure
$z$ moduli are stabilized simultaneously comparing with previous results,
showing that generically $z$ is heavier than $\tau$, and the axio-dilaton can
not be fixed first. This is also the case when we add the warping correction to
the potential; and in general this inclusion does not destabilize the moduli.
We point out why in this setup non-supersymmetric vacua are much more dense
than supersymmetric vacua. We study the dependence of the vacua stabilization
by addition of an $\overline{D3}$-brane, for a fixed volume. The vacuum state
masses have divergent eigenvalues near the conifold singularity that gets
softened by the warping correction. Generically the considered contributions
allow to find stable vacua in this setup. The scale hierarchy between 6D and
4D; arising from the warping, as stated by Giddings Kachru and Polchinski
(GKP), is preserved; while the masses in the vacua acquire physically
reasonable values.
