Wild Ramification and Stacky Curves

The local structure of Deligne–Mumford stacks has been studied for decades, but most results require a tameness hypothesis that avoids certain phenomena in positive characteristic. We tackle this problem directly and classify stacky curves in characteristic p > 0 with cyclic stabilizers of order p using higher ramification data. Our approach replaces the local root stack structure of a tame stacky curve, similar to the local structure of a complex orbifold curve, with a more sensitive structure called an Artin–Schreier root stack, allowing us to incorporate the ramification data directly into the stack. A complete classification of the local structure of stacky curves, and more generally Deligne–Mumford stacks, will require a broader understanding of root structures, and we begin this program by introducing a higher-order version of the Artin–Schreier root stack. Finally, as an application, we compute dimensions of Riemann–Roch spaces for some examples of stacky curves in positive characteristic and suggest a program for computing spaces of modular forms using the theory of stacky modular curves.