CHAPTER 1 Preliminaries from algebraic geometry In Section 1 of this Chapter we review some standard terminology, conventions, and notation from algebraic geometry we also introduce some new terminology and notation that will prove useful throughout the book. In Section 2 we very briefly discuss categorical quotients in algebraic geometry, specifically in the context of al- gebraic varieties, algebraic stacks, affine schemes, and p—adic formal schemes. Our main purpose here is to offer a quick overview of some of the available approaches to quotients in algebraic geometry before we embark, in the next Chapter, onto the task of explaining the S—geometric approach proposed in this book. In Section 3 we discuss a class of correspondences on algebraic curves that will play a key role in our book these are the correspondences that admit an analytic uniformization and will fall into 3 categories (spherical, flat and hyperbolic). The concept of ana- lytic uniformization of a correspondence is closely connected, as we shall see, to (a generalization of) the concept of postcritical finiteness in complex dynamics. The impatient reader might choose to simply skim through this Chapter and to later come back to the material here as needed. A good idea, nevertheless, is to read (at least) the numbered new definitions in this Chapter. With regards to proofs, our general policy, in this Chapter, was to prove only the facts for which we could find no convenient reference these facts are all "easy". The facts (some of which are "hard") for which we could find references are not supplied with proofs. 1.1. Algebro-geometric terminology 1.1.1. Categories. Recall the following terminology already used in the In- troduction. DEFINITION 1.1. A correspondence in a category C is a tuple (1.1) X = ( X , A i , a 2 ) , where X, X are objects in C and cri,T2 : X —• X are morphisms in C. If a := (X, cri, cr2) then we also say that o is a correspondence on X and we write (1.2) X = ( A » . We sometimes refer to X as the base of X. A categorical quotient for X is a pair (y, 7r) where Y is an object of C and n : X Y is a morphism in C satisfying the following properties: 1) 7T O &i = 7T O (J25 2) For any pair (Y',irf) where Y' is an object of C and IT' : X Y' is a morphism such that ir' o o~l = IT' o r2 there exists a unique morphism 7 : Y Y' SUCh t h a t 7 O 7T = 7T . 3
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