Close connectedness of the moduli stack of reduced curves

We prove that the moduli stack of all reduced $n$-pointed curves is ``closely connected" in characteristic zero, in the sense that each irreducible component of the stack intersects the component of smoothable curves. We achieve this by performing a detailed study of Ishii’s territories, moduli schemes parametrizing reduced curve singularities together with a normalization map. We give explicit equations for territories, bound their dimensions, describe certain functoriality properties, and study the action of several groups on territories. Along the way, we prove the existence of nonsmoothable reduced curve singularities in new ranges, generalizing work of Mumford, Pinkham, Greuel, and Stevens.