GRASSMANNIANS THE FIRST EXAMPLE OF A MODULI SPACE 1 What is this course about Many objects of algebraic geometry such as subspaces of a linear space smooth curves of genus g or stable vector bundles on a curve themselves vary in alge braically de ned families Moduli theory studies such families of algebraic objects Roughly speaking a moduli problem is the problem of understanding a given geometrically meaningful functor from the category of schemes to sets To make this more concrete consider the following three functors Example 1 1 Example 1 The Grassmannian Functor Let S be a scheme E a vector bundle on S and k a positive integer less than the rank of E Let Gr k S E Schemes S sets be the contravariant functor that associates to an S scheme X subvector bundles of rank k of X S E Example 1 2 Example 2 The Hilbert Functor Let X S be a projective scheme O 1 a relatively ample line bundle and P a xed polynomial Let HilbP X S Schemes S sets be the contravariant functor that associates to an S scheme Y the subschemes of X S Y which are proper and at over Y and have the Hilbert polynomial P Example 1 3 Example 3 Moduli of stable curves Let Mg Schemes sets be the functor that assigns to a scheme Z the set of families up to isomorphism X Z at over Z whose bers are stable curves of genus g Each of the functors in the three examples above poses a moduli problem The rst step in the solution of such a problem is to construct a smooth projective variety proper scheme proper Deligne Mumford stack that represents the functor nely coarsely De nition 1 4 Given a contravariant functor F from schemes over S to sets we say that a scheme X F over S and an element U F F X F represents the functor nely if for every S scheme Y the map HomS Y X F F Y given by g g U F is an isomorphism 1 The best answer one can usually hope for such as in Examples 1 and 2 is that there is a scheme hopefully proper representing the functor There may not be such a scheme For instance for the functor in Example 3 there does not exist a ne moduli scheme representing the functor In such cases we represent the functor either in a di erent category or we relax the conditions that we impose on the representing scheme The most common alternatives are to work with stacks or to ask for the moduli space to only coarsely represent the functor De nition 1 5 Given a contravariant functor F from schemes over S to sets we say that a scheme X F over S coarsely represents the functor F if there is a natural transformation of functors F HomS X F such that 1 spec k F spec k HomS spec k X F is a bijection for every algebraically closed eld k 2 For any S scheme Y and any natural transformation F HomS Y there is a unique natural transformation HomS X F HomS Y such that Finding a representing scheme stack a moduli space is only the rst step of a moduli problem Usually the motivation for constructing a moduli space is to understand the objects this space parameterizes This in turn requires a good knowledge of the geometry of the moduli space Among the questions that arise about these moduli spaces are 1 Is the moduli space proper If not does it have a modular compacti cation Is the moduli space projective 2 What is the dimension of the moduli space Is it connected Is it irre ducible What are its singularities 3 What is the cohomology Chow ring of the moduli space 4 What is the Picard group of the moduli space Assuming the moduli space is projective which of the divisors are ample Which of the divisors are e ective 5 Can the moduli space be rationally parameterized What is the Kodaira dimension of the moduli space The second step of the moduli problem is answering as many of these questions as possible The focus of this course will be the second step of the moduli problem In this course we will not concentrate on the constructions of the moduli spaces We will often stop at outlining the main steps of the constructions only in so far as they help us understand the geometry We will spend most of the time talking about the explicit geometry of these moduli spaces We begin our study with the Grassmannian The Grassmannian is the scheme that represents the functor in Example 1 Grassmannians lie at the heart of moduli theory Their existence is the rst step for the proof of the existence of the Hilbert scheme Many moduli spaces in turn can be constructed using the Hilbert scheme On the other hand the Grassmannians are su ciently simple that their geometry is well understood Many of the constructions for understanding the geometry of other moduli spaces such as the moduli space of stable curves imitates the techniques used in the case of Grassmannians This motivates us to begin our exploration with the Grassmannian 2 Additional references For a more detailed introduction to moduli problems you might want to read HM Chapter 1 Section A H Lecture 21 EH Section VI and K Section I 1 2 Preliminaries about the Grassmannian Good references for this section in random order are H Lectures 6 and 16 GH I 5 and Ful2 Chapter 14 Kl2 and KL Let G k n denote the classical Grassmannian that parameterizes k dimensional linear subspaces of a xed n dimensional vector space V G k n naturally carries the structure of a smooth projective variety It is often convenient to think of G k n as the parameter space of k 1 dimensional projective linear spaces in P n 1 When we use this point of view we will denote the Grassmannian by G k 1 n 1 It is easy to give G k n the structure of an abstract variety In case V C n G k n becomes a complex manifold under this structure Given a k dimensional subspace of V we can represent it by a k n matrix Choose a basis for and write them as the row vectors of the matrix GL k acts on the left by multipli cation Two k n matrices represent the same linear space if and only if they are related by this action of GL k Since the k vectors span in the matrix repre sentation there must exist a non vanishing k k minor Suppose we look at those matrices that have a xed non vanishing k k minor We can normalize that this submatrix is the identity matrix This gives a unique representation for In this representation the remaining entries are free to vary The space of such matrices is isomorphic to Ak n k In case V Cn the transition functions are clearly holo morphic We thus obtain …