For an embedded stable curve over the real numbers we introduce a hyperplane arrangement in the tangent space of the Hilbert scheme. The connected components of its complement are labeled by embeddings of the graph of the stable curve to a compact surface. We explain how the real type of a deformation of the curve is determined by the topology of such a surface. Finally, we prove that a so-called graph curve is the special fiber of a maximal Mumford curve (MM-curve) if and only if the graph is planar.This is based on a joint work with Bernd Sturmfels and Raluca Vlad.