Geometric Invariant Theory and Moduli Spaces of Maps David J. Swinarski Balliol College Submitted in partial fulfillment of the requirements of the Master of Science by Research Trinity 2003 1 Contents 1 Introduction 1 I Geometric Invariant Theory and Moduli Spaces 2 2 The Theory of Moduli 2 3 Geometric Invariant Theory 6 4 Cohomology of Quotients 8 5 Example: Hyperelliptic Curves 9 II Four Important Moduli Spaces 12 6 M(n, d) 12 6.1 Facts about vector bundles . 12 6.2 A G.I.T. construction of M(n, d) ........................ 13 6.3 The cohomology of M(n, d) ........................... 14 7 Mg 16 7.1 Facts and definitions . 16 8 U g(n, d) 19 9 Mg(X, β) 20 10 Motivation 21 III A G.I.T. Construction of the Moduli Space of Maps 24 11 Gieseker’s Construction of Mg Reviewed 24 11.1 The Hilbert scheme . 24 11.2 The Hilbert point of a curve . 25 11.3 Defining K˜ ..................................... 26 12 Fitting Ideals and the Construction of Mg 27 r 13 Constructing Mg(P , d): General Information 29 r 14 The G.I.T. Set-up and the Numerical Criterion for Mg(P , d) 32 15 Maps from Smooth Curves Are G.I.T. Stable 36 i 16 G.I.T. Semistable Maps Are Potentially Stable 42 16.1 First properties of G.I.T. semistable maps . 43 16.2 G.I.T. semistability implies that the only singularities are nodes . 56 16.3 G.I.T. semistable curves are reduced . 69 16.4 Potential stability . 73 17 The Construction Finished 75 18 Another Linearization 82 19 Closing: Toward Mg, U g(n, d), and Beyond 85 20 Appendix I: Equivariant Cohomology 87 21 Appendix II: Intersection Cohomology 89 22 Index of Terms and Notation 94 23 Acknowledgements 95 ii 1 Introduction We begin with an introduction to the theory of moduli and geometric invariant theory (G.I.T.), giving all the important definitions and results with references to the foundational works. We also outline Kirwan’s techniques for studying the cohomology of G.I.T. quotients. The second part of this paper is a quick tour of a menagerie of four important (families of) moduli spaces. This may seem unusual considering that the primary result of this paper is the construction of a single moduli space. However it is a philosophy we wish to emphasize that lessons learned in studying the geometry and topology of the moduli spaces of vector bundles over an algebraic curve as well as the moduli spaces of curves provide valuable insight into the more recently studied moduli spaces of vector bundles over varying algebraic curves and the moduli spaces of maps, and that there is a richness in considering all these in relation to one another which does not emerge if each space is considered in and of itself. A project proposed by Kirwan, our motivation for part three, illustrates this richness. The third part is the heart of this paper. Working over the complex numbers, we construct a G.I.T. quotient J//SL(W ) and prove that it is isomorphic to the Kontsevich-Manin moduli r space of maps Mg(P , d). We indicate how this construction could in turn be used to study the moduli space of curves and perhaps even the moduli space of vector bundles over varying curves. An index of some of the terms and notation used for Part III is included. 1 Part I Geometric Invariant Theory and Moduli Spaces 2 The Theory of Moduli Often the set of geometric objects of a given type (or equivalence classes of geometric objects of a given type) can be parametrized by another geometric object. For instance, consider hyperelliptic curves, that is, compact Riemann surfaces C of genus g ≥ 2 admitting a holomorphic map C → P1 of degree two. By the Riemann-Hurwitz Formula, C must have 2g + 2 branch points in the Riemann sphere P1. Two hyperelliptic curves having the same branch locus {b1, . , b2g+2} are isomorphic since they are both isomorphic to the 2g+2 2 Y normalization of the curve y = (x − bi). In fact two hyperelliptic curves C1 and C2 i=1 are isomorphic if and only if their branch loci differ by an automorphism of P1. So an unordered set of 2g + 2 distinct points in P1 specifies an isomorphism class of hyperelliptic curves. We might therefore naively expect the set P1 (2g+2) − ∆ to classify hyperelliptic curves, where ∆ is the union of all subdiagonals and the superscript (2g + 2) denotes the symmetric product. Indeed this simple observation can be made precise (see Section 5 below), yielding a quasiprojective complex algebraic variety each of whose points corresponds to an isomorphism class of hyperelliptic curves of genus g. Just how general is this phenomenon? When can the set of equivalence classes of some type of geometric object itself be given the structure of a geometric object in a natural way? The mathematical framework developed to study such questions is the theory of moduli; classic references include [GIT] Ch. 5, [MS], and [New] Ch. 1. Moduli spaces arise naturally in the setting of classification problems. In addition to their utility for classification, moduli spaces are often interesting objects in their own right. Furthermore they give important information regarding how geometric objects fit together in families. So suppose A is the set of geometric objects we wish to classify under an equivalence relation ∼ on A. Curiosity might lead us to ask whether A/∼ can be given for instance the structure of a manifold or variety. More generally, we seek to define in a natural way on A/∼ the structure of an object of Cat, where we denote by Cat one of the following categories: 2 topological spaces, manifolds, varieties, schemes, or stacks. (In this paper all moduli spaces will be locally noetherian schemes of finite type.) To specify a moduli problem one must also provide a definition of a family X of objects in A parametrized by an object S ∈ Ob(Cat). (We will denote families by uppercase calligraphic letters.) It is often useful to define an equivalence relation ∼fam of families (though if one is content with only a coarse moduli space as defined below then this is not strictly necessary). The following formal properties must be satisfied: i. A family parametrized by a single point S = {s} corresponds to a single object Xs ∈ A. ii. The equivalence relation ∼fam on families reduces to ∼ when S = {s}. iii. For any morphism ϕ : S0 → S and any family X parametrized by S, there is an induced family ϕ∗X parametrized by S0. This operation ∗ ∗ ∗ ∗ is functorial in the sense that (ϕ1 ◦ ϕ2) = ϕ2 ◦ ϕ1 and idSX = X . ∗ ∗ Furthermore if X1 ∼fam X2 then ϕ X1 ∼fam ϕ X2. We provide two examples to illustrate these definitions: Example 2.1 A family X of projective curves parametrized by a scheme S is a flat mor- phism X → S each of whose geometric fibers Xs is a projective curve of genus g. We take the equivalence relation X1 ∼fam X2 to be algebraic isomorphism over S. Example 2.2 We define a family over a scheme S of algebraic vector bundles over a fixed nonsingular projective curve C to be a vector bundle X over C × S. A plausible candidate for ∼fam would be to say that two families are equivalent if there is a bundle isomorphism between them; however there is no fine moduli space for the moduli problem so defined (cf. for instance [New] page 148)! The following choice of equivalence relation yields a moduli problem which admits a fine moduli space if the rank and degree are coprime: We say that π ∼ ∗ X1 ∼fam X2 if there exists a line bundle L → S such that X1 = X2 ⊗π L. Suppose the underlying point set of M ∈ Ob(Cat) is in bijective correspondence with A/∼. For any base S ∈ Ob(Cat) and any family X of objects of A over S, there is a map fX : S → M s 7→ [Xs] 3 where [Xs] is the equivalence class in A/∼ which Xs represents. If our moduli space M is to be a truly “universal” object, then it ought to be compatible with every family X over every S ∈ Ob(Cat). That is, the maps fX should be Cat-morphisms for every X and S. What we are really dealing with is a contravariant functor F from Ob(Cat) to the category of sets; here F(S) is the set of equivalence classes of families parametrized by S, and the Cat-morphism F(ϕ): F(S2) → F(S) for any ϕ : S2 → S is given by pullback along ϕ. Note that for any S the map fX defines a map F (S): F(S) → Hom(S, M) F (S)([X ]) = fX . This collection of maps defines a natural transformation F : F → Hom(−, M). Definition 2.3 A coarse moduli space for a given moduli problem is an object M ∈ Ob(Cat) together with a natural transformation F : F → Hom(−,M) such that i. F ({s}) is a bijection. ii. For any object M 0 ∈ Ob(Cat) and natural transformation F 0 : F → Hom(−,M 0), there exists a unique natural transformation Ψ such that F 0 = Ψ ◦ F . If the natural transformation F is an isomorphism of functors, we say that M represents F and call M a fine moduli space. In this case the moduli space has even better properties: namely, there exists a universal family. Definition 2.4 A universal family U is a family parametrized by M such that for every ∗ family X parametrized by S there is a unique morphism ϕ : S → M such that X ∼fam ϕ U.