A closer look at the stacks of stable pointed curves

In the theory of the moduli-stacks of stable n-pointed curves, there are two fundamental functors, contraction and stabilization. These functors are constructed in Knudsen (1983) [3], , where they are used to show that the various ’s are DM-stacks. In the treatment of stabilization, there is a claim stated about the deformations of a pointed ordinary double point. This claim is not proved in Knudsen (1983) [3], . The goal of this paper is to prove this claim. We decided not to use descent, so the equations are less simple than in Knudsen (1983) [3], but the treatment is more elementary and the results seemingly more general.