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Blank PDF Document Stable log surfaces, trigonal covers, and canonical curves of genus 4 The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Han, Changho. 2019. Stable log surfaces, trigonal covers, and canonical curves of genus 4. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:42029707 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA Stable log surfaces, trigonal covers, and canonical curves of genus 4 A dissertation presented by Changho Han to The Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Mathematics Harvard University Cambridge, Massachusetts April 2019 © 2019 – Changho Han All rights reserved. Dissertation Advisor: Professor Harris Changho Han Stable log surfaces, trigonal covers, and canonical curves of genus 4 Abstract We describe a compactification of the moduli space of pairs (S, C) where S is iso- morphic to P1 × P1 and C ⊂ S is a genus 4 curve of class (3, 3). We show that the compactified moduli space is a smooth Deligne-Mumford stack with 4 boundary compo- M nents. We relate our compactification with compactifications of the moduli space 4 of genus 4 curves. In particular, we show that our space compactifies the blow-up of the M hyperelliptic locus in 4. We also relate our compactification to a compactification of H3 P1 the Hurwitz space 4 of triple coverings of by genus 4 curves. iii Page intentionally left blank iv CONTENTS Title page i Copyright ii Abstract iii Table of Contents v Acknowledgements vii 1. Introduction 1 1.1. Outline 6 1.2. Conventions 6 2. Preliminaries on the minimal model program 7 2.1. Singularities of the MMP 8 2.2. Main theorems of the MMP 12 2.3. The log canonical program 20 3. Moduli spaces of ‘almost K3’ stable log surfaces 23 3.1. Stable log surfaces 23 3.2. Families of stable log surfaces 25 3.3. The canonical covering stack and the index condition 27 3.4. The moduli stack 28 3.5. Stable log quadrics 34 4. Trigonal curves and stable log surfaces 37 4.1. Stable and unstable pairs in genus 4 43 5. Flips 47 5.1. Flipping a (−4) curve (Type I flip) 47 5.2. Flipping a (−3) curve (Type II flip) 55 6. Stable replacements of unstable pairs 63 6.1. Maroni special covers 65 6.2. Hyperelliptic covers 66 v F − F 6.3. The 3 3 case 69 F − F 6.4. The 1 1 case 73 F − F 6.5. The 3 1 case 73 6.6. Summary of stable replacements 79 7. Deformation theory 83 7.1. The Q-Gorenstein cotangent complex 83 7.2. Deformations of pairs 84 8. Geometry 89 H3( = +ε) 8.1. Comparison of X with the spaces of weighted admissible covers 4 1 6 89 8.2. The boundary locus of X 92 M 8.3. Comparison of X with 4 96 References 105 vi ACKNOWLEDGEMENTS First of all, I would like to thank my advisor, Joe Harris, for his guidance and enthu- siasm on algebraic geometry during my graduate studies. Without him, I would not be able to make a smooth transition into classical algebraic geometry (subject of this thesis) from my undergraduate background on differential geometry. Second, I am indebted to my co-author, Anand Deopurkar, for sharing the research problem that grew into this thesis. With his continued support, I was able to get into the track, and not getting lost in the pool of technical details in the subject. It has been a pleasure to share and build up ideas with him. I am grateful to Valery Alexeev, who showed continued interest and support to the thesis. His suggestions and questions helped me to expand the thesis. I am also thankful for his advice on technical details, which helped me to write up clear and concise proofs. I also would like to thank senior experts who I have met during conferences. Thank you Maksym Fedorchuk, Paul Hacking, János Kollár, Jenia Tevelev, and Chenyang Xu for enlightening discussions and suggestions. I cannot forget how my academic brothers, Anand Patel, Atanas Atanasov, Gabriel Bujokas, Francesco Cavazzani, Eric Riedl, Alex Perry, and Adrian Zahariuc, taught me during my graduate career in Harvard University. I learned a lot on various aspects of algebraic geometry from every one of you. Thank you my younger academic siblings Mitchell Lee, Gwyneth Moreland, Sung Gi Park, Geoffrey Smith, Dennis Tseng for giving me feedbacks on many versions of my talks on the topic of this thesis. Your presence helped me to improve highlighting key points of the thesis during my recent talks. To Kenny Ascher, Dori Bejleri, Kristin DeVleming, Patricio Gallardo, Ross Goluboff, Gio- vanni Inchiostro, and Luca Shaffler, thank you for sharing your insights and viewpoints with me on the area on moduli of surfaces during various conferences. vii I owe special thanks to my family, who continuously cheered me up and encouraged me to continue on mathematics. Special thanks to my parents who taught me to never give up and move forward. Finally, many thanks to National Sciences and Engineering Research Council of Canada (NSERC) for granting me the award [PGSD3-487436-2016]. The financial support helped me to focus on mathematical research. Thanks to you, I can always feel pride in being a Canadian. viii 1. INTRODUCTION Algebraic geometry originated in the study of zero sets of polynomials. An important question in algebraic geometry is about the classification of projective varieties. It is remarkable that certain collections of projective varieties are themselves parametrized by points of algebraic varieties, called moduli spaces. A well-known example is the M moduli space g of smooth projective curves of a fixed genus g. In general, a point of a moduli space M corresponds to an object in the classification, and a map from a space V to M corresponds to a family of objects parametrized by V . In this way, M not only gives ways to understand individual objects, but also naturally gives ways to understand families of objects. A crucial step in the study of moduli spaces is the notion of a modular compactification. A moduli space M is a modular compactification of a moduli space M if M is compact and contains M as a dense open subset. Since understanding compact spaces is much easier than noncompact spaces, it is useful to have a modular compactification. Note that moduli spaces of smooth varieties usually are not compact. To compactify them, one can enlarge the class of smooth varieties by allowing singularities. If we carefully select singularity conditions on varieties, then we often get a compact moduli space. For example, enlarging the class of smooth projective curves by adding stable curves of ≥ [ ] M genus g 2, Deligne and Mumford in 11 constructed a compact moduli space g of M M stable curves as a modular compactification of g . The discovery of g led to significant advances not only in algebraic geometry, but also in related fields such as number theory and mathematical physics. For instance, Harris and Mumford showed in [21] that the M ≥ M moduli space g is of general type for g 24 by using divisors on g , Kontsevich used M intersection theory on g to give an algebraic formulation of Gromov-Witten theory (see [16] for a survey), and Deligne and Mumford gave an algebraic proof in [11] of M M the irreducibility of g in any positive characteristics, by using the existence of g . 1 In contrast to the geometry of the moduli spaces of curves, the geometry of moduli spaces of surfaces are generally unknown. Even constructing a compact moduli space of surfaces is a recent result by Kollár, Shepherd-Barron, and Alexeev [31, 6] (see [29] for a survey). Unlike the case of curves, we do not know the boundary members for compact moduli spaces of surfaces. In fact, the geometry of moduli spaces of surfaces satisfies "Murphy’s law", i.e. it can get arbitrarily bad. The broad goal of the work is to identify and describe compact moduli spaces of surfaces which are well-behaved. In particular, the goal of this thesis is to describe a compact moduli space X that lies at the cusp of three different areas of study of moduli spaces in algebraic geometry, namely (1) the study of compact moduli spaces of surfaces of log general type, (2) the study of the birational geometry of the moduli space of curves, and (3) the study of alternative compactifications of Hurwitz spaces parametrizing branched coverings. ∼ The moduli space X is defined as follows. Consider a pair (S, D), where S = P1 × P1 and D ⊂ S is a smooth divisor of class (3, 3). Observe that for all w > 2=3, the pair (S, wD) is a surface of log general type. Set w = 2=3 + ε, where 0 < ε 1. Then X is a compactification of the space of pairs (S, wD) constructed by Kollár–Shepherd- Barron/Alexeev [31, 6] (the idea of taking ε close to 0 is inspired by the work of Hacking [20]). The compactification X parametrizes stable semi log canonical pairs of log general type. We recall the definition in detail in the main text. For now, it suffices to say that it M is the analogue in higher dimensions of the Deligne–Mumford compactification g,n.
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