Online Archive of University of Virginia Scholarship
Wild Ramification and Stacky Curves914 views
Author
Kobin, Andrew, Mathematics - Graduate School of Arts and Sciences, University of Virginia
Advisors
Obus, Andrew, Mathematics, Baruch College
Abstract
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.
Degree
PHD (Doctor of Philosophy)
Keywords
Stacks; Curves; Wild Ramification
Language
English
Rights
All rights reserved (no additional license for public reuse)
Kobin, Andrew. Wild Ramification and Stacky Curves. University of Virginia, Mathematics - Graduate School of Arts and Sciences, PHD (Doctor of Philosophy), 2020-04-24, https://doi.org/10.18130/v3-y9s6-kt47.