  In a suitable regime of superstring theory, D-branes in a Calabi-Yau space
and their most fundamental behaviors can be nicely described mathematically
through morphisms from Azumaya spaces with a fundamental module to that
Calabi-Yau space. In the earlier work [L-L-S-Y] (D(2): arXiv:0809.2121
[math.AG], with Si Li and Ruifang Song) from the project, we explored this
notion for the case of D1-branes (i.e. D-strings) and laid down some basic
ingredients toward understanding the notion of D-string world-sheet instantons
in this context. In this continuation, D(10), of D(2), we move on to construct
a moduli stack of semistable morphisms from Azumaya nodal curves with a
fundamental module to a projective Calabi-Yau 3-fold $Y$. In this Part I of the
note, D(10.1), we define the notion of twisted central charge $Z$ for
Fourier-Mukai transforms of dimension 1 and width [0] from nodal curves and the
associated stability condition on such transforms and prove that for a given
compact stack of nodal curves $C_{\cal M}/{\cal M}$, the stack
$FM^{1,[0];Zss}_{C_{\cal M}/{\cal M}}(Y,c)$ of $Z$-semistable Fourier-Mukai
transforms of dimension 1 and width [0] from nodal curves in the family
$C_{\cal M}/{\cal M}$ to $Y$ of fixed twisted central charge $c$ is compact.
For the application in the sequel D(10.2), $C_{\cal M}/{\cal M}$ will contain
$C_{\bar{\cal M}_g}/\bar{\cal M}_g$ as a substack and $FM^{1,[0];Zss}_{C_{\cal
M}/{\cal M}}(Y,c)$ in this case will play a key role in defining stability
condition for morphisms from arbitrary Azumaya nodal curves (with the
underlying nodal curves not necessary in the family $C_{\cal M}/{\cal M}$) to
$Y$.
