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Moduli Spaces of (N1 1 Preliminaries You should know the basics of category theory, schemes, varieties, morphisms, proper, flat, Hilbert Polynomial, and the genus of a curve. References: Harris and Morrison "Moduli of Curves" (Ch 1,xA), Harris "Al- gebraic Geometry" (Lecture 21), Eisenbud and Harris "Geometry of Schemes" (Section VI), Kollar "Rational Curves on Algebraic Varieties" General Overview: People want to classify objects, so pick a set of properties and attempt to make the set of objects of that sort into a variety. We want to say carefully "What is a moduli problem?" and w"What does it mean that a particular variety (scheme, stack) solves this moduli problem?" The answer consists of two parts. 1. What is M? 2. What can we say about the geometry of M? For part 2, these are the types of questions we ask about M: 1. Is the moduli space proper? If not, does it have a modular compactifica- tion? Is the moduli space projective? 2. What is the dimension? Is the moduli space connected? Is M irreducible? What kinds of singularities does it have? 3. What is the cohomology ring/Chow ring of the moduli space? 4. What is the Picard group of M? If M is projective, can one describe the ample divisors? The effective divisors? 5. Can the moduli space be rationally parametrized? What is its Kodaira dimension? What makes a moduli problem? 1. A collection A of algebro-geometric objects (a) For a fixed variety or scheme X, let A be the collection of configura- tions of n distinct points on X. (b) A collection of smooth curves of genus g (c) Morphisms P1 ! Pn (d) Hypersurfaces of degree d in Pn. 2. An equivalence relation ∼ on A with M the underlying set of points of A= ∼ and the geometry of M reflecting how objects move in families. (a) ∼ can be trivial or if X = Pr, A= ∼ can be the configurations of points up to projective equivalence (b) ∼ is up to isomorphism. 1 (c) ∼ is up to isomorphism of maps (commuting diagrams as follows:) ' n .... n P ............................................................................................................. P . ... ........ ........ ...... ..... ..... ..... ..... g ..... ..... ..... f . ..... ..... ..... ..... ..... ..... ..... P1 (d) ∼ is up to projective transformation 3. Notion of an equivalence class of families of A= ∼. (a) If X is a scheme and A is a configuration of n distinct points on X with ∼ relation, then an equivalence class of families is an equivalence class of diagrams B × X !π1 B with n sections B ! B × X such that −1 for b 2 B closed point, π1 (b) = b × X ' X and so σ1(b); : : : ; σn(b) gives n distinct points of X. (b) A family parametrized by B of smooth curves of genus g up to iso- morphism is a flat morphism X !π B where for each b 2 B, π−1(b) is an isomorphism class of smooth curves of genus g. (c) A family of isomorphism classes of morphisms f : P1 ! Pr is a µ .... r X ............................................................................................................. P . π . ....... ... diagram B for each b 2 B a closed point, π−1(b) ' P1 1 r and µjπ−1(b) : P ! P (d) A family of hypersurfaces of degree d in Pr is a diagram ... r X ............. ............. ................ ........... B × P .... ..... ..... ..... ..... ..... ..... ..... ..... ..... π ..... ..... ..... π1 . ..... ..... ..... ..... ..... ...... ... ........ B −1 r such that for all b 2 B closed points, π (b) = Hb ! b × P is a hypersurface of degree d We will now look at part II, families, in more depth: Definition 1.1 (Family of Objects). Let A be a collection of algebro-geometric objects and ∼ an equivalence relation on A. A family of objects of A= ∼ parametrized by a scheme (or variety) B is a morphism π : X ! B satisfy- ing three properties: 1. If B = Spec(k), then X consists of a single element of A= ∼ 2 2. We can define an equivalence relation ∼ on X ! B which restricts to the original equivalence relation if B = Spec(k) 3. Families pull back to families functorially: if π : X ! B and f : B0 ! B, then ∗ 0 .... f X = B ×B X .................................... X . ........ ........ ... f ... 0 ............................................................................................................. B .... B This pull back operation satisfies the following: (a) (f ◦ f 0)∗X = (f 0)∗f ∗X ∗ 0 (b) If the family is idB : B ! B, then we get f B = B and the pullback ∗ family is idB0 : f B ! B. (c) If X ! B and X0 ! B are families and X ∼ X0, then f ∗X ∼ f ∗X0. We will fix some notation: If X ! B is a family, we will write f ∗X = 0 B ×B X = XB0 , so if we have b ! B an inclusion of a point, then Xb is the fiber of the family over b 2 B. Suppose that M is a scheme whose underlying set of points is A= ∼. Then if X ! B is a family of elements of A= ∼, we get a "classifying map" ηX : B ! M which will take a closed point b to [Xb]. If M is any sort of moduli space, then at minimum we require that this map be a morphism. Ideally ηX should define a bijective correspondence between equivalence classes of families X ! B and morphisms B ! M . We begin by defining a contravariant functor F : fSchemesg ! fSetsg by B 7! F (B) = fequivalence classes of families parameterized by Bg. If we have f 2 Mor(B; B0), then F (B) ⊂ F (B0) and take the morphism to be X ! B maps to f ∗X ! B0. We want to say what M (a scheme whose underlying points are A= ∼) has to satisfy in order to be the answer to the problem posed by this functor F . We consider the functor of points hom(∗; M ): fschemesg ! fsetsg which takes X to hom(X; M ). We define φ : F ! hom(∗; M ) by putting B 2 Obj(fSchg), φ(B): F (B) ! hom(B; M ) by X ! B is sent to ηX : B ! M. If f 2 f η Mor(B; B0) for B; B0 schemes, we get a map B0 ! B !X M which we can compose to get a map B0 ! M . We say that M solves the problem posed by F if ' is a natural isomorphism, that is, (M ; φ) represents F . Definition 1.2 (Fine Moduli Space). A fine moduli space for a given moduli problem described by a functor F is the pair (M ; φ) that represents F . Notice: 1. φ(Spec k): F (Spec k) = A= ∼→ Mor(Spec k; M ) ' M is a bijection. 3 2. φ(M ): F (M ) ! Mor(M ; M ). The latter contains idM which corre- sponds to a unique family U ! M (the Universal Family), such that any family is a pullback of this family. Why is this? If X ! B is sent to ηX id ∗ B ! M ! M composes to id ◦ηX , so X ' (id ◦ηX ) U . Definition 1.3 (Fine Moduli Space (alternate)). A fine moduli space consists of a scheme (or variety or stack) M and a family U ! M called the uni- versal family such that for every X ! B there exists a unique morphism (the ∗ classifying morphism) ηX : B ! M for which X = ηX U Example 1.1. Take X = P1, and F (P1; n) that we are looking for is the moduli space of configurations of n distinct points on P1 with the trivial equivalence relation. F (P1; n) = P1×:::×P1n[diagonals, where there are n P1's and the diagonals are the subloci where points coincide. Let U = F (P1; n)×P1 with map π : U ! 1 F (P ; n) the first projection. It comes with n sections σi where σi = pi ◦id where 1 1 1 th 1 pi : P × ::: P n ∆ ! P projects to the i copy of P . 2 Lecture 2 X ' P1 (more generally, a scheme over S and sometimes even a stack) Agenda: F (X; n) for n 2 N is a fine moduli space that was studied by topologists originally (n points on X), and it was given a compactification by Fulton and MacPherson, X[n]. But for now, we will look at G(k; n), the Grassmanian, which we will use ¯ to study Chow Varieties G(k; d; n), which will be used to construct M0;n, the moduli space of n-pointed stable curves of genus zero. Recall that solving a moduli problem has two stages, we are going to discuss when we can expect to have a moduli space and when they can be constructed. Definition 2.1 (Fine Moduli Space). Let F be a contravariant functor F : fSchemes=Sg ! fSetsg we say a scheme X(F ) and U (F ) 2 F (X(F )) repre- sents the functor finely if for every scheme Y the map φ(Y ) : hom(Y; X(F )) ! ∗ ... g U (F ) ..................................................................... U (F ) . ........ ........ ... g ... ... F (Y ) given by g : Y ! X(F ) maps to the square Y ................................................................................................ X(F )is an isomorphism. Note, there simply may not be such a pair (X(F ); U (F )). A rule of thumb for evaluating whether a fine moduli space exists is that if E 2 A= ∼ has Aut(E) 6= id, then there is no chance. Why? Suppose that there is an E with nontrivial automorphism group. Then one can use Aut(E) to define a nontrivial family π : X ! B such that 4 every fiber X0 ' E. If there were a fine moduli space X(F ) for families of A= ∼, then we could show that X ! B is trivial. Indeed: If (X(F ); U (F )) is a fine moduli space, then X ' ηX U (F ), where ηX : B ! X(F ) is the defining map. That is, X ' B ×pt U (F ), which is trivial, and so assuming such a nontrivial family can be constructed, there can be no such moduli space.
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