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Wild Ramification and Stacky Curves

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Wild Ramification and Stacky Curves Wild Ramification and Stacky Curves Andrew Kobin Cedarburg, Wisconsin Bachelor of Science, Wake Forest University, 2013 Master of Arts, Wake Forest University, 2015 A Dissertation Presented to the Graduate Faculty of the University of Virginia in Candidacy for the Degree of Doctor of Philosophy Department of Mathematics University of Virginia May, 2020 ©Copyright by Andrew Kobin 2020 All Rights Reserved Abstract The local structure of Deligne–Mumford stacks has been studied for decades, but most results require a tameness hypothesis that avoids certain phenom- ena 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 struc- ture of a tame stacky curve, similar to the local structure of a complex orb- ifold curve, with a more sensitive structure called an Artin–Schreier root stack, allowing us to incorporate the ramification data directly into the stack. A com- plete classification of the local structure of stacky curves, and more generally Deligne–Mumford stacks, will require a broader understanding of root struc- tures, 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 char- acteristic and suggest a program for computing spaces of modular forms using the theory of stacky modular curves. Acknowledgments I always knew this would be the last part of my thesis I would write – even in the final days of my journey, there are still people supporting me and helping me across the finish line. What I mean to say is, I would not be here without the generosity and encouragement of a long list of people, too many of whom I am leaving out here. I first want to heap praise upon my parents, Jean and Peter, who taught me nearly everything I know. I am sure they will steadfastly deny any credit and redirect it back to me, but they are incredible people and I owe them a lot. My brothers, Danny and Billy, are likewise tireless sources of support and inspiration for me. I love how much you all dove into this with me, even though the math might have seemed daunting! My adviser, Andrew Obus, has been a huge influence on this project and on me as a young mathematician. I struggled mightily with some of the math in the chapters that follow, but I cannot recall feeling for a single moment that I was incapable of finding my way through and that level of confidence was due to Andrew’s unwavering encouragement. I specifically want to thank him for his time helping me edit this thesis, which is, shall we say, not a quick read. I would like to thank several people who contributed in meaningful ways to the development of this project, including: David Rydh, Vaidehee Thatte, John Voight and David Zureick-Brown. Thank you also to the scores of people who listened to me spill my ideas on the floor and play around with them, including: John Berman, Brandon Boggess, Thomas Brazelton, Soumya Sankar and Libby Taylor. The University of Virginia math department has been the perfect home the i last five years and there are many people to thank for my success here. Fac- ulty members such as Julie Bergner, Paul Bourdon, Francesco di Plinio, Valia Gazaki, Craig Huneke, Nick Kuhn, Sara Maloni, Tom Mark, Tai Melcher, An- drei Rapinchuk, Jim Rolf, Charlotte Ure and Lloyd West were all incredibly generous with their time and advice. The administrative staff – Allison Boese, Raven James, Zvezdana Kish and Robin Koontz Mongold – are simply won- derful people to interact with and made the department run smoothly at all times. And I am fortunate to have met so many extraordinary graduate stu- dents who have passed through the department over the years and enriched my life. Beyond the walls of Kerchof Hall, the city of Charlottesville has also earned a place in my heart. I have made some dear friends here who I want to ac- knowledge for being great human beings, cheering me on when I was excited about something, and sympathizing when I was down. To Chris Chung, Matt Feller, Zach Gates, Eloísa Grifo, Peter Johnson, Bogdan Krstic, Chris Leonard, Kristy Mangold, Jim Phillips, Mark Schrecengost, George Seelinger and Mo- jdeh Tarighat: I love you all. To the talented musicians of University Records: thanks for letting me get some musical energy out with y’all! Finally, to B: I love you and am grateful for your support these last five years. It has not been an easy time, but I wouldn’t be here without you. ii Contents 0 Introduction1 0.1 Motivating Problem..........................1 0.2 Structure of the Thesis........................4 0.3 Connections to Other Works.....................6 1 Ramification8 1.1 Discretely Valued Fields.......................8 1.2 Local Fields.............................. 14 1.3 Ramification Theory......................... 19 1.4 Cyclic Extensions........................... 24 1.4.1 Kummer Theory....................... 26 1.4.2 Artin–Schreier Theory.................... 29 1.4.3 Artin–Schreier–Witt Theory................. 34 2 Some Algebraic Geometry 44 2.1 Scheme Theory............................ 45 2.1.1 Affine Schemes........................ 45 2.1.2 Schemes............................ 47 2.1.3 Sheaves of Modules..................... 54 iii 2.1.4 Group Schemes........................ 61 2.1.5 Étale Morphisms....................... 65 2.2 Curves................................. 69 2.2.1 Divisors............................ 71 2.2.2 The Riemann–Roch Theorem................ 77 2.2.3 The Riemann–Hurwitz Formula.............. 81 2.2.4 Artin–Schreier Theory for Curves............. 83 3 Algebraic Stacks 86 3.1 Sites................................... 87 3.1.1 Grothendieck Topologies and Sites............. 87 3.1.2 Sheaves on Sites........................ 91 3.1.3 Cohomology......................... 96 3.1.4 Cechˇ Cohomology...................... 100 3.1.5 Direct and Inverse Image.................. 102 3.1.6 The Étale Site......................... 105 3.2 Categories Fibred in Groupoids................... 116 3.2.1 Fibred Categories....................... 121 3.2.2 Fibre Products......................... 128 3.3 Descent................................. 133 3.3.1 Galois Descent........................ 134 3.3.2 Faithfully Flat Descent.................... 139 3.3.3 Effective Descent Data.................... 141 3.3.4 Stacks............................. 143 3.4 Algebraic Spaces........................... 150 3.4.1 Definitions and Properties.................. 151 iv 3.4.2 Sheaves on Algebraic Spaces................ 155 3.5 Algebraic Stacks............................ 157 3.5.1 Definitions and Properties.................. 158 3.5.2 Deligne–Mumford Stacks.................. 167 3.5.3 Sheaves on Deligne–Mumford Stacks........... 170 3.5.4 Coarse Moduli Spaces.................... 173 4 Stacky Curves 177 4.1 Basics.................................. 177 4.1.1 Definitions........................... 177 4.1.2 Divisors and Vector Bundles................ 179 4.1.3 Line Bundles and Sections.................. 186 4.2 Tame Stacky Curves......................... 189 4.2.1 Root Stacks.......................... 190 4.2.2 Classification Results..................... 197 4.3 Wild Stacky Curves: The Z=pZ Case................ 198 4.3.1 Artin–Schreier Root Stacks................. 198 4.3.2 Classification Results..................... 205 4.4 Wild Stacky Curves: The Higher-Order Cyclic Case....... 218 4.4.1 Garuti’s Compactification.................. 219 4.4.2 Artin–Schreier–Witt Root Stacks.............. 225 5 Applications 229 5.1 Canonical Rings............................ 229 5.1.1 The Tame Case........................ 231 5.1.2 The Wild Case......................... 232 v 5.2 Modular Forms............................ 240 5.2.1 Background.......................... 241 5.2.2 Geometric Modular Forms................. 247 5.2.3 Modular Forms in Characteristic p ............. 256 vi Chapter 0 Introduction 0.1 Motivating Problem Over the complex numbers, Deligne–Mumford stacks with generically triv- ial stabilizer can be understood by studying their underlying complex orbifold structure. This approach leads one to the following classification of stacky curves: over C, or indeed any algebraically closed field of characteristic 0, a smooth stacky curve is uniquely determined up to isomorphism by its under- lying complex curve and a finite list of numbers corresponding to the orders of the (always cyclic) stabilizer groups of the stacky points of the curve (cf. Sec- tion 4.2.2 and [GS]). This key feature of stacky curves in characteristic 0 paves the way for many applications, including the description of the canonical ring of a tame stacky curve in [VZB] (cf. Section 5.1) and a stacky approach to com- puting rings of modular forms (cf. 5.2 and [VZB, Ch. 6]). Such a concise classification of stacky curves fails in positive characteristic since stabilizer groups may be nonabelian. Even in the case of stacky curves 1 over an algebraically closed field k of characteristic p > 0, there exist infinitely many nonisomorphic stacky curves over k with the same underlying scheme and stacky point with stabilizer group abstractly isomorphic to Z=pZ (cf. Re- mark 4.3.17). Thus any attempt at classifying stacky curves in characteristic p will require finer invariants than the order of the stabilizer group. Our main results, stated below, provide a classification of stacky curves in characteristic p > 0 having nontrivial stabilizers of order p, using higher ramification data at the stacky points. -1 A new tool called an Artin–Schreier root stack, denoted }m ((L, s, f)=X), will be constructed and studied in Section 4.3.1, but it appears in the statements of -1 our main results below so a brief introduction is in order. The object }m ((L, s, f)=X) is defined using the data of a line bundle L on X and two sections s 2 H0(X, L) pr and f 2 H0(X, L⊗m). It replaces the root stack (L, s)=X from the theory of tame stacky curves. Our main results are the following theorems.
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