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In a suitable regime of superstring theory, D-branes in a Calabi-Yau spaceand their most fundamental behaviors can be nicely described mathematicallythrough morphisms from Azumaya spaces with a fundamental module to thatCalabi-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 thisnotion for the case of D1-branes (i.e. D-strings) and laid down some basicingredients toward understanding the notion of D-string world-sheet instantonsin this context. In this continuation, D(10), of D(2), we move on to constructa moduli stack of semistable morphisms from Azumaya nodal curves with afundamental module to a projective Calabi-Yau 3-fold $Y$. In this Part I of thenote, D(10.1), we define the notion of twisted central charge $Z$ forFourier-Mukai transforms of dimension 1 and width [0] from nodal curves and theassociated stability condition on such transforms and prove that for a givencompact 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-Mukaitransforms 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 $FM^{1,[0];Zss}_{C_{\calM}/{\cal M}}(Y,c)$ in this case will play a key role in defining stabilitycondition for morphisms from arbitrary Azumaya nodal curves (with theunderlying nodal curves not necessary in the family $C_{\cal M}/{\cal M}$) to$Y$.

A mathematical theory of D-string world-sheet instantons, I: Compactness  of the stack of $Z$-semistable Fourier-Mukai transforms from a compact family  of nodal curves to a projective Calabi-Yau 3-fold