Higher Dimensional Trichotomy Conjectures in Model Theory by Benjamin Castle A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Thomas Scanlon, Chair Associate Professor Pierre Simon Associate Professor Adityanand Guntuboyina Spring 2021 Higher Dimensional Trichotomy Conjectures in Model Theory Copyright 2021 by Benjamin Castle 1 Abstract Higher Dimensional Trichotomy Conjectures in Model Theory by Benjamin Castle Doctor of Philosophy in Mathematics University of California, Berkeley Professor Thomas Scanlon, Chair In this thesis we study the Restricted Trichotomy Conjectures for algebraically closed and o-minimal fields. These conjectures predict a classification of all sufficiently complex, that is, non-locally modular, strongly minimal structures which can be interpreted from such fields. Such problems have been historically divided into `lower dimensional' and `higher dimen- sional' cases; this thesis is devoted to a number of partial results in the higher dimensional cases. In particular, in ACF0 and over o-minimal fields, we prove that all higher dimensional strongly minimal structures whose definable sets satisfy certain geometric restrictions are locally modular. We also make progress toward verifying these geometric restrictions in any counterexample. Finally, in the last chapter we give a full proof of local modularity for strongly minimal expansions of higher dimensional groups in ACF0. i To Katrina and Justin ii Contents Contents ii 1 Introduction 1 1.1 Historical Background . 1 1.2 The Challenge of Working in Higher Dimensions . 3 1.3 Outline of Results . 5 2 Preliminaries on Strongly Minimal Sets 7 2.1 Strong Minimality and Dimension . 7 2.2 Pregeometries and Local Modularity . 13 2.3 Plane Curves . 20 2.4 Trichotomy Conjectures and Our Setting . 26 3 First Observations and Almost Purity 29 3.1 Dimension and Genericity . 29 3.2 Pure Parts of Sets . 35 3.3 Almost Purity and Almost Closedness . 38 3.4 Relation to Covering Maps . 40 4 Finite Fundamental Groups and Unimodularity 46 4.1 Review of Unimodularity . 47 4.2 Proof I: Reducing to Unimodularity . 51 4.3 Proof II: Producing a Pairwise Disjoint Family . 57 4.4 A Non-Unimodular Example with Trivial Geometry . 67 5 The General Case of Smooth Varieties 73 5.1 Algebro-Geometric Preliminaries . 74 5.2 Relative Almost Purity . 76 5.3 Relative Almost Purity of the Fiber Product of a Family . 78 5.4 Proof of the Main Theorem . 84 5.5 Interpreting a Group . 89 6 Unimodularity Revisited: An O-minimal Variant 95 iii 6.1 O-Minimality: Review of Basic Definitions and Facts . 96 6.2 Some Algebraic Topology . 99 6.3 Our Setting . 105 6.4 Dimension and Genericity . 106 6.5 Geometric Preliminaries . 108 6.6 Almost Purity . 112 6.7 Coverings . 113 6.8 Components . 117 6.9 The Main Theorem . 125 7 Toward Guaranteeing Almost Purity 128 7.1 Common and Semi-indistinguishable Points . 129 7.2 Sweeping Sets . 135 7.3 Generic Almost Closedness . 138 7.4 Frontiers of Plane Curves in General . 150 7.5 Toward Showing Almost Closedness Implies Almost Purity . 155 8 The Case of Groups 174 8.1 Standard Reductions . 175 8.2 Almost Closedness and Abelian Varieties . 179 8.3 Frontiers of Generic 2-Hypersurfaces . 179 8.4 Frontiers of Arbitrary 2-Hypersurfaces . 193 8.5 Poles . 196 8.6 The Main Theorem . 199 Bibliography 201 iv Acknowledgments I would first like to thank my advisor, Tom Scanlon, for mentoring me throughout my time at Berkeley. I feel very lucky to have been one of his students, and this thesis would not have been possible without his endless patience, support, and mathematical wisdom. I also thank the other members of the committee, Pierre Simon and Adityanand Gun- tuboyina, for their time and effort. I especially thank Pierre for his contributions to the model theory group at Berkeley, including running our seminar. Similarly, I am thankful to Carol Wood and Ward Henson for their presence in the Berkeley logic community. This thesis was inspired by the work of too many mathematicians to name. However, I would like to mention Assaf Hasson and Dmitry Sustretov, whose ideas formed the starting point for this project; and especially Assaf, for a number of helpful conversations on the material. Next, I thank all of my peers within the logic community at Berkeley, for making graduate school a fun and memorable experience. These include, but are by no means limited to, fellow model theorists Nick Ramsey, Reid Dale, Mariana Vicaria, Adele Padgett, David Casey, Diego Bejarano Rayo, Scott Mutchnik, and Sebastian Eterovi´c. I also thank my former housemates, Ian Gleason, Ravi Fernando, and Patrick Lutz, for a myriad of stimulating conversations, both within and outside of mathematics; I enjoyed developing friendships with each of them during my time at Berkeley. I especially thank Ravi for patiently fielding a number of probably silly questions from a model theorist trying to do algebraic geometry; and Ian for being my best friend for the last six years. Thank you as well to all those who mentored me mathematically before my Berkeley years: in particular, Adrienne Stanley, Douglas Mupasiri, and TJ Hitchman, my undergraduate professors; P´eterJuh´asz,my teacher, mentor, and friend from my time in Budapest; and Brian Reynolds, my high school math coach. Last but not least, I thank my entire family { in particular my parents, and my brother Tommy { for their unwavering support. My parents recognized my love of math early on, and I wouldn't have made it to this point without their constant encouragement. I especially thank my wife, Katrina, for supporting me in so many ways, and helping me to stay sane during these last few months of writing. And finally, I thank my five month old son, Justin. Much of this thesis was typed with him sitting or sleeping in my lap; and even though he did everything he could to keep me fully distracted the whole time, having him around truly makes pushing through the hardest moments worth it. 1 Chapter 1 Introduction 1.1 Historical Background The class of strongly minimal sets has long been of importance to Model Theory, originating in 1971 with its use in Baldwin and Lachlan's treatment of Morley's Categoricity Theorem [2]. The study of such sets, however, found a new and exciting avenue in the work of Boris Zilber in the 1970s and 1980s, which proposed a classification according to familiar algebraic objects. Zilber (see, for example, [53] and [54]) established a trichotomy which divided strongly minimal sets according to the complexity of their induced strongly minimal structures. He proceeded to classify those of `middle' complexity { the non-trivial locally modular sets { as those whose `geometries' arise from linear algebra over division rings (a more precise form of this identification was given later by Hrushovski [23] { see Fact 2.4.2 of this thesis). Following this line of thought, Zilber conjectured that the most complex level { the non-locally modular strongly minimal sets { should be classifiable as those arising from algebraic geometry over algebraically closed fields. Precisely, he showed that non-locally modular strongly minimal sets are exactly those which contain certain combinatorial objects called `pseudoplanes' (see [53], Theorem 3.1 or [54], Chapter II, Theorem 3.5); he then conjectured the following (see [53], Conjecture B): Conjecture 1.1.1 (Zilber's Trichotomy Conjecture). Every uncountably categorical pseu- doplane is mutually interpretable with an algebraically closed field. In particular, every non- locally modular strongly minimal set interprets an algebraically closed field. This conjecture has an enticing flavor: indeed, it would suggest that algebraically closed fields are characterizable (up to mutual interpretability) according to purely model theoretic principles. Alas, Zilber's conjecture turned out to be false, as proven in 1993 by Ehud Hrushovski [21]: Theorem 1.1.2 (Hrushovski Construction). There is a strongly minimal set which is not locally modular and does not interpret any infinite group, thus in particular cannot interpret an algebraically closed field. CHAPTER 1. INTRODUCTION 2 Nevertheless, Zilber's original prediction has retained significant importance to model theory { in particular, by guiding much of the development of geometric stability theory over the last 35 years. Since Hrushovski's refutation of Zilber's original conjecture, model theorists have proceeded to discover a multitude of instances in which similar trichotomy phenomena do hold. These instances include, for example: strongly minimal sets definable in differentially closed fields (see [26] and [41]). strongly minimal sets which carry a `Zariski-like' topology [27]. o-minimal structures viewed locally near a point [34]. In each of the above examples, a classification is achieved along Zilber's original lines, dividing the specified objects into the `trivial,' `vector space,' and ‘field’ cases. Thus it has long been apparent that various geometric situations in Model Theory witness the same surprising pattern: the emergence `out of thin air' of the same familiar algebraic objects (vector spaces and fields). The common guiding principle which fuels these classification results is the presence of an underlying geometry on definable sets. For example, one may choose to work only with definable sets that are `closed' in a meaningful sense. A more difficult challenge, then, is to establish trichotomy phenomena without such an available notion of geometry. Of course, as noted above, the trichotomy fails in full generality even for strongly minimal sets. However, an interesting `in between' question is to ask about structures for which (i) there is a geometry controlling the definable sets, but (ii) that geometry can only be seen from a certain `outside' object, not necessarily definable from the structure itself.