The moduli stack of $A_r$-stable curves

This paper is the first in a series of four papers aiming to describe the (almost integral) Chow ring of ${\bar{\mathcal{M}}}_3$, the moduli stack of stable curves of genus $3$. In this paper, we introduce the moduli stack ${\tilde{\mathcal{M}}}_{g,n}^r$ of $n$-pointed $A_r$-stable curves and extend some classical results about ${\bar{\mathcal{M}}}_{g,n}$ to ${\tilde{\mathcal{M}}}_{g,n}^r$, namely the existence of the contraction morphism. Moreover, we describe the normalization of the locally closed substack of ${\tilde{\mathcal{M}}}_{g,n}^r$ parametrizing curves with $A_h$-singularities for a fixed $h\le r$.