Stability conditions and infinitesimal deformation of curves

Let $\mathcal{X}$ be an infinitesimal deformation of a smooth projective curve $X_0$ over a field. We study stability conditions under such deformations and show that the derived push-forward functor associated with the inclusion $X_0\to \mathcal{X}$ induces an isomorphism between the space of stability conditions on $\mathcal{X}$ and that on $X_0$. This yields a direct comparison between the deformed and undeformed settings. As an application, we prove that the autoequivalence group $\mathrm{A}\mathrm{u}\mathrm{t}{\mathbf{D}}^b(\mathcal{X})$ naturally acts on ${\mathbf{D}}^b(X_0)$, providing a link between derived symmetries and the deformation structure.