R. M. Schwarz

Gromov-Witten invariants of the classifying stack of principal Gm-bundles

Master’s thesis

Supervisor: Dr. D. Holmes

Date exam: 24 August 2018

Mathematisch Instituut, Universiteit Leiden Contents

Introduction 2

1 The stack BG 4 1.1 Algebraic stacks ...... 4 1.2 Group schemes ...... 7 1.3 Torsors and principal bundles ...... 9 1.4 The algebraic stack [X/G]...... 10 1.5 The stack BGm as stack of line bundles ...... 13

2 Quasi-coherent sheaves on BG 16 2.1 Representations and G-equivariant quasi-coherent sheaves . . 16 2.2 Quasi-coherent sheaves on BG ...... 18 2.3 Quasi-coherent sheaves on BGm ...... 22

3 Moduli stack of Gieseker bundles 25 3.1 Stable curves ...... 25 3.2 Stack of principal bundles ...... 27 3.3 Gieseker bundles ...... 28 3.4 Stack of Gieseker Gm-bundles ...... 31

4 Gromov-Witten invariants 36 4.1 Evaluation maps ...... 36 4.2 Gromov-Witten invariants ...... 40 4.3 Admissible line bundles ...... 42 4.4 Coherent pushforward of an admissible complex ...... 46 4.5 Gromov-Witten invariants for Me 0,3(BGm)...... 50

References 52

1 Introduction

In algebraic geometry, moduli spaces are studied to answer questions related to classification problems, such as classifying lines in the plane through the origin by the projective line. In this thesis, we are interested in

Me g,I (BGm), which is a moduli stack for Gieseker Gm-bundles, which are line bundles satisfying certain conditions, on (modifications of) stable curves of genus g marked by an ordered set I. This is the stack to consider if we want to study line bundles on stable curves. However, it is not a Deligne-Mumford algebraic stack, but an Artin algebraic stack. In the article [6], the authors attempt to study Gromov-Witten invariants of this stack. In particular, they prove coherence of the pushforwards of certain specific K-theory classes along the forgetful map

F : Me g,I (BGm) → Mg,I to the stack of stable curves of genus g marked by I. The aim of this thesis is to give examples to illustrate the article [6] and prove its main results in the case where the genus is 0 and we only have three marks.

Overview In chapter one, the classifying stack BG of G-bundles for a group scheme G is defined, for which we introduce the concepts of stacks, group schemes, bundles and torsors. In particular we are interested in BGm for the mul- tiplicative group scheme Gm, in which case BGm will be the stack of line bundles, considered with isomorphisms. In chapter two, a proof is given of the fact that for a smooth group scheme G over a field k, there is an equivalence of categories

{Quasi-coherent sheaves on the stack BG} ↔ {Representations of G}.

In particular, if G = Gm we can classify the coherent sheaves on BGm by studying the finite-dimensional representations of Gm. In the third chapter, we introduce the moduli stack of Gieseker bundles Me g,I (BGm), and explain by examples why we study Gieseker Gm-bundles instead of the stack of principal Gm-bundles. In chapter four, we turn to the main theorem in [6], introducing first the natural evaluation maps that we study for the Gromov-Witten invariants and then the admissible line bundles and admissible complexes of coherent

2 sheaves. Each concept is illustrated for the case of genus 0 curves with 3 marks. Finally, we show how the pushforwards of these admissible complexes along the forgetful map

F : Me 0,3(BGm) → M0,3 where M0,3 is a point, are actually coherent sheaves. Moreover, we compute the numerical Gromov-Witten invariant attached to the pushforward of such an admissible complex.

Preliminaries The readers of this thesis should have a background in algebraic geometry, so they may for example be master students with an algebraic geometry specialisation. We assume that the reader is familiar with some scheme theory, and is familiar with category theory, because we shall be working with stacks. For T a scheme, we may use the letter T for the scheme T , but also for the category SchT , although this should not lead to confusion. Schemes are generally considered with the ´etaletopology, so a collection of maps

{fi : Xi → X} for schemes Xi,X is a covering if each fi is ´etaleand the fi are jointly surjective.

Acknowledgements I would like to thank my advisor dr. David Holmes for his time and support during the entire project. I would also like to thank my fellow student Heleen Otten for her support, and PhD student Mark van den Bergh MSc for his help in the writing process. Finally I would like to thank my family and Aslan, for occasionally creating the purrfect study environment.

3 1 The stack BG

The purpose of this chapter is to define the classifying stack of G-bundles for a group scheme G, and then deduce that for the multiplicative group scheme Gm, BGm can be described as the stack of line bundles.

1.1 Algebraic stacks Firstly we need to introduce stacks and an algebraic stacks. In [1], one finds a short introduction to algebraic stacks, although we prefer to use the more thorough book [2] as reference. To define a stack, we need the concept of descent, which we introduce here because it will also be important for studying quasi-coherent sheaves on the stack BGm. We use the notation found in [2] paragraph 4.2. Definition 1.1. Let C be a category and let p: F → C be a fibered category over C, then we write F (Y ) for the fiber (category) over an object Y in C, meaning all objects in F that map to object Y in C and all morphisms mapping to idY .

Definition 1.2. A category fibered in groupoids over a category C is a fibered category p: F → C such that, for each object Y in C, the fiber category F (Y ) is a groupoid, i.e., a category where all morphisms are iso- morphisms.

Definition 1.3. Let C be a category with finite fiber products. Consider a fibered category p: F → C, and a morphism f : X → Y in C and an object E in F (X). A descent datum σ for an object E is an isomorphism ∗ ∗ σ : pr1E → pr2E in F (X ×Y X) satisfying the following compatibility or co- cycle condition in F (X ×Y X ×Y X). Write pri : X ×Y X → X for the projec- tion to the i-th component and prij : X×Y X×Y X → X×Y X for the projec- tion to the i-th and j-th component, then we have canonical isomorphisms ∗ ∗ ∼ ∗ ∗ ∗ ∗ ∼ ∗ ∗ ∗ ∗ ∼ ∗ ∗ pr12pr2E = pr23pr1E and pr12pr1E = pr13pr1E and pr13pr2E = pr23pr2E. We require

∗ ∗ ∗ pr12σ ∗ ∗ ∼ ∗ ∗ pr12pr1E pr12pr2E pr23pr1E

∗ ∼ pr23σ ∗ ∗ ∗ pr13σ ∗ ∗ ∼ ∗ ∗ pr13pr1E pr13pr2E pr23pr2E to commute.

4 Definition 1.4. Let C be a category with finite fiber products, p: F → C be a fibered category and f : X → Y a morphism in C. We define the category F (f : X → Y ) by having objects pairs (E, σ), where E ∈ F (X) is an object and σ is a descent datum for E. A morphism (E0, σ0) → (E, σ) in F (X → Y ) is a 0 ∗ ∗ 0 morphism g : E → E in F (X) such that σ ◦ pr1g = pr2g ◦ σ .

More generally, if {Xi → Y }i∈I is a set of morphisms in C, we de- fine F ({Xi → Y }i∈I ) to be the following category: the objects are data ∗ ∗ ({Ei}i∈I , {σij}i,j∈I ), where Ei ∈ F (Xi) and σij : pr1Ei → pr2Ej is an iso- morphism in F (Xi ×Y Xj) for each i, j ∈ I satisfying the desired com- patibility conditions in F (Xi ×Y Xj ×Y Xk). Again we refer to the set of isomorphisms {σij}i,j∈I as the descent data on the {Ei}i∈I . Note that for the collection of morphisms {fi : Xi → Y }, there is a functor

: F (Y ) → F ({Xi → Y }),

∗ sending an object E to the data ({fi E}, {σcan,ij}) where σcan is the canonical isomorphism.

Definition 1.5. For an object ({Ei}i∈I , {σij}i,j∈I ) ∈ F ({Xi → Y }i∈I ) as above, we say the descent data {σij}i,j∈I are effective if ({Ei}i∈I , {σij}i,j∈I ) is in the essential image of : F (Y ) → F ({Xi → Y }).

Definition 1.6. A category fibered in groupoids over a site C, p: F → C, is a stack if and only if

(i) For any object X in C and objects x, y in the fiber F (X), the presheaf Isom(x, y):(C/X)op → Set defined by

∗ ∗ Isom(x, y)(f : Y → X) := IsomF (Y )(f x, f y)

is a sheaf.

(ii) For any covering {Xi → X}i∈I of an object X in C, any descent data with respect to this covering are effective.

Equivalently to (i) and (ii), we may require that for every object X in C and covering {Xi → X} the functor : F (X) → F ({Xi → X}) is an equivalence of categories.

5 A morphism of stacks p: F → C, q : G → C is simply a morphism of categories over C, that is, a functor a: F → G such that q ◦ a = p.

Most stacks in this thesis will be stacks over the site C = SchS for S a scheme with the ´etaletopology, such as BSG over SchS in Example 1.25 or Me 0,3(BGm) over Schk in Definition 3.12. Definition 1.7. Let f : X → Y and g : Z → Y be morphisms of categories

fibered in groupoids over category C, then the fiber product X ×Y Z has as objects triples (x, z, α) where x and z are objects of X and Z respectively and α: f(x) → g(z) is an isomorphism in Y. A morphism (x0, z0, α0) → (x, z, α) is a pair of isomorphisms (a: x0 → x, b: z0 → z) is such that g(b)◦α0 = α◦f(a) as maps f(x0) → g(z). The fiber product of stacks is again a stack, by 4.6.4 in [2]. By 8.1.16, the fiber product of algebraic stacks, which will be defined in Definition 1.14, is also an algebraic stack. Definition 1.8. Let S be a scheme and consider the category of S-schemes with the ´etaletopology. Let P be a property of morphisms of schemes which is satisfied by isomorphisms and closed under composition. We say property P is stable with respect to the ´etaletopology if for all morphisms f : Z → Y of S-schemes and covers {Yi → Y }, f has P if and only if all the fi : X ×Y Yi → Yi have P . From Proposition 5.1.4 in [2], we get the following result. Lemma 1.9. The following properties are stable: surjective, proper, flat, ´etale and smooth. Recall that a functor F : Cop → Set is representable if it is naturally op isomorphic to hX : C → Set,Y 7→ HomC (Y,X) for some object X in C. Definition 1.10. Let S be a scheme and f : F → G a morphism of sheaves on SchS with the ´etaletopology. Then f is representable by schemes if, for every S-scheme T and morphism of sheaves hT → G, the fiber product F ×G hT is representable. Definition 1.11. Let P be a stable property of morphisms of schemes and let f : F → G be a morphism of sheaves on SchS with the ´etaletopology. If f is representable by schemes, we say that f has property P if, for every

S-scheme T and morphism hT → G, the morphism of schemes induced by pr1 : hT ×G F → hT has P . Note that there is also a way to define, for example, a smooth morphism of stacks that is not representable. However, we will not need that definition.

6 Definition 1.12. Let S be a scheme. An algebraic space over S is a functor op X : SchS → Set such that • X is a sheaf with respect to the big ´etaletopology,

• the diagonal ∆: X → X ×S X is representable by schemes, and

• there exists an S-scheme U and a surjective ´etalemorphism hU → X.

Morphisms of algebraic spaces over S are morphisms of functors.

Note that the second condition in the definition, as shown in Lemma

5.1.9 in [2], implies that any morphism hT → X for T a scheme and X an algebraic space is representable by schemes. Therefore, in the third condition, hU → X is representable and it makes sense to require it to be ´etaleand surjective.

Definition 1.13. A morphism of stacks f : X → Y is representable if, for every scheme T and morphism T → Y, the fiber product X ×Y T is an algebraic space.

Lemma 8.1.3 in [2] then gives that, if a morphism of stacks X → Y is representable, then the fiber product X ×Y X is an algebraic space, not just for schemes but for every algebraic space X and morphism X → Y.

Definition 1.14. A stack X over SchS is an algebraic stack in the sense of Artin (resp. Deligne-Mumford) if

• the diagonal ∆: X → X ×S X is representable, and

• there is a smooth (resp. ´etale)and surjective morphism of stacks

SchT → X for a scheme T . A morphism of algebraic stacks is a morphism of stacks.

An example of an Artin algebraic stack, is given by BG in Example 1.25 and Proposition 1.26.

1.2 Group schemes Definition 1.15. A group scheme G over a scheme S is a group object in the category of schemes over S. Equivalently, a group scheme over S is a scheme G over S together with a op factorisation of the functor hG : SchS → Set over Grp.

7 op Hom(−,G) SchS Set

forget Grp

Example 1.16. The multiplicative group Gm,S represents the functor

op ∗ ∗ SchS → Grp,T 7→ Γ(T, OT ) = OT (T )

∗ where Γ(T, OT ) is the multiplicative group of invertible elements in OT (T ). −1 As a scheme, we have Gm,S = SpecS(OS[X,X ]).

We can illustrate this when S = Spec(R) is affine. Then Gm,S equals −1 ∼ 1 SpecS(R[X,X ]) = AS \{0} and we factorise HomS(−, Gm,S) over Grp by the group structure for each scheme T over S specified via

∼ −1 ∼ ∗ HomS(T, Gm,S) → HomR−alg(R[X,X ], OT (T )) → OT (T ) .

When working over a field k, we often write Gm instead of Gm,k. The following examples are relevant for representation theory for group schemes.

Example 1.17. For a field k and n ∈ Z>0, the general linear group GLn,k is the affine group scheme

Spec(k[Xij,Y ]i,j=1,...,n/(det(Xij)Y − 1)), where det(Xij) is the determinant formula in the variables Xij (seeing the Xij as coefficients in an n by n matrix). Again we explain the name by considering T = Spec(R) an affine k- scheme. Then we have

Hom (T, GL ) ∼ Hom (k[X , det(X )−1],R). Schk n,k = k−alg ij ij

Hence this is equivalent to give n2 elements of R, namely the images of the

Xij, which can be viewed as coefficients in an n by n matrix, in such a way that the determinant is invertible. Thus we assign to T = Spec(R) the multiplicative group of invertible n by n matrices. Note that, by definition, GL1,k = Gm,k = Gm. We can reformulate this example to the following.

8 Example 1.18. Let V be a finite-dimensional k-vector space. Then we naturally view V as quasi-coherent sheaf on Schk via associating to t: T → k the OT -module V ⊗k OT . We define Autk(V ) as group scheme to be

op Autk(V ): Schk → Grp, ∗ (t: T → k) 7→ AutT (VT ) = AutT (t V ) = AutT (V ⊗k OT ), where we mean automorphisms as the quasi-coherent sheaf over T . A mor- 0 ∗ 0∗ ∗ phism f : T → T over k is mapped to AutT (t V ) → AutT 0 (t V ), σ 7→ f σ. Note that if V is infinite-dimensional, we can still consider the above functor Autk(V ), but it need not be representable.

1.3 Torsors and principal bundles Firstly we introduce torsors, as in 4.5.1. in [2]. Definition 1.19. Let C be a site and let µ be a sheaf of groups on C.A µ-torsor on C is a sheaf P on C together with a left action ρ: µ × P → P such that

(1) for every object X in C there exists a covering {Xi → X} such that P(Xi) 6= ∅ for all i, and

(2) the map µ × P → P × P, (g, p) 7→ (p, gp) is an isomorphism. Equivalently to the second condition we may require that if P(X) is non- empty, then µ(X) acts on P(X) simply transitively. A µ-torsor (P, ρ) is trivial if P has a global section. Then we have an isomorphism µ → P, g 7→ gp identifying P and µ and ρ with left translation. A morphism of µ-torsors (P, ρ) → (P0, ρ0) is a morphism f : P → P0 of sheaves such that ρ µ × P P

idµ ×f f ρ0 µ × P0 P0 commutes. ∗ In the article [6] the following definition of a principal C -bundle is used: ∗ ∗ “principal C -bundle on a scheme X is a scheme P on which C acts freely ∗ from the right and a C -invariant map p: P → X which is locally trivial: X ∼ ∗ ∗ has an open cover {Ui} such that Ui ×X P = Ui × C , C -equivariantly.” Slightly generalised to fit Gm as group scheme over k, we use the defini- tion found in 4.5.4 in [2].

9 Definition 1.20. Let G be a flat locally finitely presented group scheme over scheme X.A principal G-bundle over X is a pair (π : P → X, ρ) where π is a flat locally finitely presented surjective morphism of schemes and

ρ: G ×X P → P an action such that the map

(ρ, pr2): G ×X P → P ×X P is an isomorphism. A morphism of principal G-bundles (P, ρ) → (P 0, ρ0) is a morphism of X-schemes f : P → P 0 such that the following diagram commutes:

idG ×f 0 G ×X P G ×X P

ρ ρ0 f P P 0. There is a connection between torsors and principal G-bundles. Fix a scheme X and let G be a flat locally finitely presented X-group scheme and let µ = hG = HomX (−,G) be the sheaf of groups. Given a principal G-bundle (P, ρ), we can define the µ-torsor (P, ρ˜) by setting

P = HomX (−,P ): SchX → Set with action induced by ρ:

HomX (T,G) × HomX (T,P ) → HomX (T,P ), (f, g) 7→ ρ ◦ (f × g) for a X-scheme T . This construction defines a fully faithful functor from principal G-bundles on X to µ-torsors on SchX with the fppf topology. In specific cases, this is also an equivalence of categories, as stated in [2] Proposition 4.5.6.

Lemma 1.21. If the structure morphism G → X is affine, then the functor from principal G-bundles on X to µ-torsors on SchX associating HomX (−,P ) to a principal G-bundle (P, ρ), is an equivalence of categories.

Remark 1.22. In the case where G is a smooth affine group scheme, such as G = Gm, Remark 4.5.7 in [2] states that the category of principal G-bundles on X is equivalent to µ-torsors on SchX with the ´etaletopology. Also, if we had defined a principal G-bundle over X as an algebraic space over X satisfying the above properties, we have an equivalence between principal G-bundles and µ-torsors.

1.4 The algebraic stack [X/G] We define [X/G] as in Example 8.1.12 in [2].

10 Definition 1.23. Let X be a scheme over S and let G be a smooth group scheme over S which acts on X from the left. We define [X/G] to be the stack with objects triples (T, P, π) where

• T is an S-scheme,

•P is a GT = G ×S T -torsor on SchT with the ´etaletopology, and

• π : P → X ×S T is a GT -equivariant morphism of sheaves on SchT , i.e., a morphism of GT -torsors.

A morphism (T 0, P0, π0) → (T, P, π) is a pair (f, f b) where f : T 0 → T is a b 0 ∗ morphism of S-schemes and f : P → f P is an isomorphism of GT -torsors on SchT 0 such that the diagram

f b P0 f ∗P π0 f ∗π 0 X ×S T commutes.

Here we used the notation GT and X ×S T also for the sheaf represented by GT and X ×S T respectively at points 2 and 3. Also, implicitly we used that if G is a group scheme over S, then GT is a group scheme over T . By ∼ definition of the fiber product, we have HomT (Y,GT ) = HomS(Y,G) and so we give HomT (Y,GT ) the group structure inherited from HomS(Y,G). We consider [X/G] as a category fibered in groupoids over SchS via the functor [X/G] → SchS, (T, P, π) 7→ T . Because of descent theory for sheaves, we also have that [X/G] is actually a stack. The proof that it is an (Artin) algebraic stack when G is smooth, can be found in the example of [2]. Here we prove it for our main example, Example 1.25.

Remark 1.24. It is important to note the following about maps from an S- scheme T to [X/G]: because [X/G](T ) is the category of triples (T, P, π) in

[X/G], giving a map SchT → [X/G] is equivalent to specifying a GT -torsor P on SchT and a GT -equivariant π : P → X ×S T . Example 1.25. Let S be a base scheme and G a smooth group scheme over

S. The map G ×S S → S as morphism over S must be pr2 and so the action is trivial. We describe the algebraic stack [S/G], also known as BG or BSG. Let T be an S-scheme, then giving a G ×k T - torsor on SchT is equivalent to giving a G-torsor on SchT . We need not specify the morphism of torsors ∼ π : P → S ×S T = T over T , because it must be the trivial map. Therefore

11 objects of BG are pairs (T, P) of an S-scheme T and a G-torsor P on SchT . Morphisms (T 0, P0) → (T, P) then are pairs (f, f b) where f : T 0 → T and f b : P0 → f ∗P is an isomorphism of torsors over T 0. By Remark 1.24, maps from any S-scheme T to BG are specified by giving a G-torsor on T . By Remark 1.22, if we assume G is affine or if we work with principal G-bundle that are algebraic spaces, we have that giving a G-torsor on T is equivalent to giving a principal G-bundle on T . Proposition 1.26. Let S be a base scheme and G a smooth group scheme over S with trivial action on S. The stack BG is an Artin algebraic stack. Proof. Consider the morphism of stacks f : S → BG defined by the trivial bundle sending t: T → S to (T,G ×S T ). Given a morphism of k-schemes 0 b 0 0 t: T → T , we get a morphism (t, t ):(T,G ×k T ) → (T ,G ×k T ) because b ∗ 0 there is a natural isomorphism of G-bundles t : G ×k T → f (G ×k T ) = 0 G ×k T ×T 0 T . This is a morphism of stacks over the category SchS. Let X be any S-scheme X with a map X → BG specified by a principal ∗ G-bundle E, so FE : X → BG, (a: T → X) 7→ (T, a E). To deduce that f is representable, consider X ×BG S. By definition of the fiber product of stacks, for a k-scheme T we have ∼ (X×BGS)(T ) = {(a, b, φ) | a ∈ SchX (T ), b ∈ SchS(T ), and φ: f(b) → FE (a)} ∼ ∼ ∗ = {(a, φ) | a: T → X, and an isomorphism φ: G ×S T = a E}. Because the isomorphism φ must be an isomorphism of G-bundles on T , so a map that respects the G-action, it is enough to know the image of {e} × T ∗ where e is the unit. So equivalently, we may give a map s: T → a E = T ×X E of G-bundles over T . By property of the fiber product, we conclude ∼ ∼ X ×BG S(T ) = {(a, s) | a: T → X, s: T → T ×X E} = HomX (T, E).

Therefore, X ×BG S is representable, by E, for all such (X, E). So the morphism f is representable by a scheme. Also, X ×BG S → X, or as schemes E → X, is indeed surjective by definition of a G-bundle and smooth because if G/k is smooth, then any principal G-bundle is smooth. Therefore, f : S → BG is a surjective, smooth, representable morphism.

The diagonal ∆: BG → BG ×S BG is representable if and only if for all

S-schemes T and maps FP1 × FP2 : T → BG ×S BG specified by G-torsors P1, P2, the sheaf 0 ∗ ∗ Isom(P1, P2): SchT → Set, (t: T → T ) 7→ IsomT 0 (t P1, t P2) mapping to the set of isomorphisms of torsors over T 0, is actually an algebraic space, because

12 Isom(P1, P2) T

FP1 ×FP2 ∆ BG BG ×S BG

is a pullback diagram. To verify that Isom(P1, P2) is an algebraic space, we may consider an ´etalecover U → T by a scheme U and verify that

Isom(P1, P2) pulled back to U is an algebraic space. Therefore we may assume that P1, P2 are trivial torsors, so isomorphic to GT , and we have

0 ∗ ∗ Isom(P1, P2)(t: T → T ) = Isom(t GT , t GT ) = Isom(GT 0 ,GT 0 ).

0 0 0 Note that giving an isomorphism G ×S T → G ×S T over T as G-bundles 0 is equivalent to giving g ∈ G(T ). So Isom(P1, P2) is representable by a scheme when P1 and P2 are trivial, and so it is an algebraic space and so the diagonal is representable. Hence BG is an Artin algebraic stack.

This proof also shows that BG is the classifying stack for principal G- bundles: any principal G-bundle p: P → X is pulled back by a unique morphism φ: X → BG (defined by bundle P) from the universal principal G-bundle f : S → BG: p P X φ (1) f S BG.

1.5 The stack BGm as stack of line bundles

We want to identify the stack BGm with the stack of line bundles.

Definition 1.27. A line bundle on a scheme X is a locally free OX - module of rank 1. Equivalently, a line bundle on a scheme X is a coherent sheaf L on X such ∼ that there is a cover {Ui → X}i with L|Ui = OUi .

Also common is to define a line bundle on X as an invertible sheaf on X, where we mean that it is an invertible element in the monoid consisting of coherent sheaves on X with the operation ⊗OX and unit OX . Example 1.28. As a simple and useful example, we introduce the line bun- 1 dles O(n) on Pk for a field k. The notes of [4] (chapter 14) provide an in- 1 troduction to line bundles, including this example. Let Pk = Proj(k[x0, x1]) and we describe the sheaf O(n) by defining it to be trivial on the standard opens U0 = D(x0) = Spec(k[x1/x0]) and U1 = D(x1) = Spec(k[x0/x1]). We

13 glue by the transition function U0 → U1, which is defined by multiplying n −n by (x0/x1) = (x1/x0) , and U1 → U0, which is defined by multiplying by n −n (x1/x0) = (x0/x1) . These line bundles satisfy the following properties: O(n) = O(1)⊗n and O(m) ⊗ O(n) = O(m + n). Also, we can explicitly compute the global sections. Such a global section are polynomials f(x1/x0) ∈ k[x1/x0] and g(x0/x1) ∈ k[x0/x1] such that

−1 n f((x0/x1) )(x0/x1) = g(x0/x1) and therefore there are no global sections if n < 0. Moreover, f must have degree lesser than or equal to n if n ≥ 0. Therefore, we have

0 1 dim H (Pk, O(n)) = n + 1 if n ≥ 0. The invertible sheaves up to isomorphism form an abelian group under the 1 ∼ tensor product, called the Picard group. We have Pic(Pk) = Z and every 1 invertible sheaf on Pk is of the form O(n) for some n ∈ Z. We define the degree of the line bundle to be this n.

Let X be a scheme and consider C = SchX with the ´etaletopology. ∗ Write O : SchX → CRing, {f : Y → X} 7→ f OX (Y ) = OY (Y ) for the natural sheaf of commutative rings, then an invertible sheaf on C is a sheaf of O-modules L such that for any object U in SchX , there exists a covering {Ui → U}i∈I such that the restriction of L to the localized site C/Ui, so to SchUi , is isomorphic to the restriction of O viewed as O-module. Recall ∗ that Gm,X = Gm : SchX → CRing, {f : Y → X} 7→ OY (Y ) and that giving a Gm,X - torsor on SchX is the same as a Gm-torsor on SchX . Proposition 1.29. There is an equivalence of categories

{Gm − torsors on X} ↔ {line bundles on X}, where the latter category is only considered with isomorphisms.

Proof. Let X be a scheme and L a line bundle on X, so an invertible OX - module. We define the associated Gm-torsor PL of L op PL : SchX → Set by sending an object f : U → X in SchX to the set of isomorphisms of O| - modules O| → L| . There is a natural action of on SchU SchU SchU Gm ∗ PL making PL in a Gm-torsor. An element of a ∈ OU (U) induces an isomorphisma ˜ of O| and then we define SchU (U) × Isom(O| , L| ) → Isom(O| , L| ), (a, φ) 7→ φ ◦ a.˜ Gm SchU SchU SchU SchU

14 Checking the two conditions for a torsor, firstly choose a cover {Ui → U} where L|Ui is trivial and there we have ∼ ∼ ∗ PL(Ui) = Isom(O|Sch , L|Sch ) = Isom(O|Sch , O|Sch ) = OUi (Ui) Ui Ui Ui Ui which is non-empty for all non-empty Ui. Secondly, whenever PL(U) is non- ∗ empty, the OX (U) action is simply transitive, because locally on trivial ∗ ∗ opens, the elements of OX (U) precisely differ by elements of OX (U) . Let L1, L2 be line bundles and let φ: L1 → L2 be an isomorphism of line bundles, then we have the map PL1 → PL2 defined on U ∈ SchX by Isom(O| , L | ) → Isom(O| , L | ), f 7→ φ ◦ f, SchU 1 SchU SchU 2 SchU where φ ◦ f is again an isomorphism because φ is. This morphism indeed respects the Gm-action by associativity of the composition. op To construct an inverse, let X be a scheme and P : SchX → Set be a Gm-torsor. Define a Gm-action on the product OX × P by ∗ −1 OX (U) × (OX (U) × P(U)) → OX (U) × P(U), (a, (s, t)) 7→ (sa , at).

∗ Consider the quotient OX \OX × P: as presheaves we simply mean ∗ ∗ OX \OX × P(U) = OX (U) \ (OX (U) × P(U)) and then we sheafify to obtain LP . Because P is a Gm-torsor we can choose a U (in an appropiate ´etalecover of X) such that T (U) is non-empty, and we can choose t ∈ T (U). Then we know that because of the simply transitive ∗ ∼ action that OX (U) → T (U), a 7→ at is an isomorphism. Then we have ∗ ∼ OX (U) \ (OX (U) × P(U)) = OX (U). Therefore, as a presheaf, LP is indeed ´etalelocally trivial. Using that a Gm-torsor for the ´etale topology is trivial also Zariski locally (this is known for GLn-torsors, see for example [14]), we can also deduce that LP is trivial Zariski locally as a presheaf. However the isomorphism also gives an isomorphism on stalks and so also the sheaf LP is locally isomorphic to OX . A morphism of torsors φ : P1 → P2 is sent to the map induced by id ×φ on OX × P1. These constructions give an equivalence of categories.

Hence, given a Gm-torsor P on a scheme X, there is an associated line bundle, and given a line bundle L on X, there is an associated Gm-torsor. The trivial principal Gm-bundle Gm ×k T corresponds to the Gm-torsor ∼ ∼ ∗ HomT (−, Gm×kT ) = Homk(−, Gm) = OT , which in turn via the proposition corresponds to the trivial line bundle ∗ ∗ ∼ OT \OT × OT = OT .

15 2 Quasi-coherent sheaves on BG

Schemes in this section are considered with the ´etaletopology. In chapter 4, we study pullbacks of coherent sheaves on BG, and so it is useful to classify quasi-coherent sheaves on BG. We will prove the following.

Theorem 2.1. Let G be a smooth group scheme over k. There is an equiv- alence of categories

{Quasi-coherent sheaves on BG} ↔ {Representations of G}.

2.1 Representations and G-equivariant quasi-coherent sheaves Firstly, we define a representation of a group scheme over field k, using

Definition 1.18 of Autk(V ).

Definition 2.2. A representation of a group scheme G over a field k is an morphism of functors G → Autk(V ) for V a k-vector space.

The category of representations of group scheme G over k, denoted as Repr (G), has as objects representations, so pairs (V, φ) of a k-vector space k V and a representation φ: G → Autk(V ). Morphisms (V, φ) → (W, ψ) are G-equivariant maps h: V → W , i.e., a k-linear map such that for all g ∈ G(T ), the diagram

φ(g) VT VT

hT hT ψ(g) WT WT commutes, where hT is the pullback of h via T → pt. We use the following category as intermediate step to prove the equiva- lence in theorem 2.1.

Definition 2.3. Let S be a scheme and G a smooth group scheme over S and X a scheme over S on which G acts via ρ: G×S X → X.A G-equivariant quasi-coherent sheaf on X is a pair (F, σ) where F is a quasi-coherent sheaf ∗ ∼ ∗ on X and σ : ρ F → pr2F is an isomorphism such that for any S-scheme T 0 and g, g ∈ G(T ), the diagram of quasi-coherent sheaves on T ×S X

ρ∗σ ∗ ∗ ∗ g g0 ∗ ∗ ρgρg0 (pr2F ) ρg(pr2F )

∼ σg σ ∗ ∗ g0g ∗ ρg0g(pr2F ) pr2F

16 commutes, where

• pr2 : T ×S X → X is the projection,

• ρg is pr1 × (ρ ◦ (g × idX )): T ×S X → T ×S X the map induced by the action, and ∗ ∗ ∗ • σg : ρgpr2F → pr2F is the pullback of σ via map g × idX .

Lemma 2.4. Let X,Y be S-schemes with G-actions aX : G ×S X → X and aY : G ×S Y → Y respectively and let f : Y → X be a G-equivariant morphism of schemes, that is, f ◦ aY = aX ◦ (idG ×f). Let (F, σ) be a ∗ ∗ G-equivariant quasi-coherent sheaf on X, then (f F, (idG ×f) σ) is a G- equivariant quasi-coherent sheaf on Y .

∗ ∗ Proof. Firstly, f F is a quasi-coherent sheaf on Y . Then (idG ×f) σ is an isomorphism from

∗ ∗ ∼ ∗ ∗ ∼ ∗ ∗ (idG ×f) aX F = (aX ◦ (idG ×f)) F = (f ◦ aY ) F = aY (f F ) to

∗ ∗ ∼ ∗ ∗ ∼ ∗ ∗ (idG ×f) prX F = (prX ◦ (idG ×f)) F = (f ◦ prY ) F = prY (f F ).

For any S-scheme T and g, g0 ∈ G(T ) the commutativity of the diagram of quasi-coherent sheaves on T ×S Y follows directly from pulling back the commutative diagram on T ×S X via (idT ×f).

Denote the category with objects quasi-coherent G-equivariant sheaves on a scheme X by QcohG(X). A morphism (F, σ) → (E, τ) is a morphism f : F → E of quasi-coherent sheaves on X such that

ρ∗F σ pr∗F

ρ∗f pr∗f ρ∗E τ pr∗E commutes. The next proposition is the reason for introducing G-equivariant quasi-coherent sheaves.

Proposition 2.5. Let G be a smooth group scheme over k and let pt = Schk. There is an equivalence of categories

QcohG(pt) ↔ Repr (G). k

17 Proof. Write p: G → pt for the trivial map. Let (F, σ) be an object of G ∗ Qcoh (pt), then we can see F as a k-vector space and σ ∈ AutG(p F ) defines a scheme morphism ρσ : G → Autk(F ) sending a g ∈ G(T ) to ∗ ∗ ∗ ∼ ∗ g σ ∈ AutT (g p F ) = AutT (t F ) where T is t: T → k. This is a group ho- momorphism: G-equivariance structure gives that σg0g = σgσg0 and also σe is idempotent and an isomorphism, so the identity. Then ρσ : G → Autk(F ) is a representation.

Given a representation ρ: G → Autk(V ), the k-vector space V natu- rally gives a quasi-coherent sheaf on pt. Also,ρ ˜ := ρ(idG) is an element of ∗ ∗ AutG(p V ) such that G(T ) → AutT (VT ), g 7→ ρ(g) = g ρ˜ is a group homo- morphism:ρ ˜ satisfies (g0g)∗ρ˜ = g∗ρg˜ 0∗ρ˜ for g, g0 ∈ G(T ). Therefore (V, ρ˜) is a G-equivariant sheaf on pt. A morphism in Repr (G) also corresponds to a morphism in QcohG(pt); k let f :(F, σ) → (E, τ) be a morphism in QcohG(pt), and consider the rep- resentations ρσ : G → Autk(F ) and ρτ : G → Autk(E). The induced linear map f : F → E by definition satisfies τ ◦ p∗f = p∗f ◦ σ, and then for all k-schemes T and for all g ∈ G(T ) we have by pulling back along g that

τg ◦ fT = fT ◦ σg, and so we indeed have a morphism of representations. For the other direction, given a map h: V → W between representa- tions φ: G → Autk(V ) and ψ : G → Autk(W ), it induces a morphism (V, φ(idG)) → (W, ψ(idG)). Because h satisfies hT ◦ φ(g) = ψ(g) ◦ hT for ∗ ∗ all g ∈ G(T ), for g = idG we get the desired p h ◦ φ(idG) = ψ(idG) ◦ p h. The constructions are inverses, so QcohG(pt) is equivalent to Repr (G). k

2.2 Quasi-coherent sheaves on BG At this point it is important to specify what we mean by a quasi-coherent sheaf on a stack. We follow the construction of Olsson, [2].

Definition 2.6. Let X/S be an algebraic stack, that is a stack X over the category SchS. Define Sch/X as the category containing as objects pairs (T, t) where T is a scheme over S and t: T → X is a morphism of stacks over S. The morphisms (T 0, t0) → (T, t) in Sch/X are pairs (f, f b) where f : T 0 → T is a morphism of S-schemes and f b : t0 → t ◦ f is an isomorphism of functors T 0 → X. Consider the site Lis-Et(´ X), the lisse-´etalesite on X, which is the full subcategory of Sch/X consisting of the pairs (T, t) where t: T → X is a b smooth morphism, and a covering is a collection {(fi, fi ):(Ti, ti) → (T, t)} such that {fi : Ti → T } is an ´etalecover of T .

18 Remark 2.7. Note that for each (T, t) ∈ Lis-Et(´ X), we have an inclusion from the ´etalesite on T to the lisse-´etalesite on X, Et(´ T ) → Lis-Et(´ X) given by sending h: T 0 → T to (T 0, t ◦ h). Then a functor F : Lis-Et(´ X)op → Set is a sheaf if and only if for all (T, t) the restriction of F to Et(´ T ), denoted by F(T,t), is a sheaf. ´ Definition 2.8. We define the sheaf OX on Lis-Et(X) by associating OT (T ) to (T, t).

Remark 2.9. By construction 9.1.10 in Olsson [2], we may also describe ´ a sheaf of OX-modules on Lis-Et(X) by the following data: for each ob- ´ ject (T, t) ∈ Lis-Et(X), an ´etalesheaf F(T,t) of O-modules on T , and for each morphism (f, f b):(T 0, t0) → (T, t) in Lis-Et(´ X) a morphism of sheaves ∗ φ(f,f b) : f F(T,t) → F(T 0,t0) satisfying two conditions. Firstly, the φ(f,f b) need ∗ to be compatible with composition, that is φ(f,f b)◦(g,gb) = φ(g,gb) ◦ g φ(f,f b). b Secondly, if we have (f, f ) with f ´etale,the map φ(f,f b) has to be an iso- morphism. We may describe a morphism of sheaves

{F(T,t), φ(f,f b)} → {E(T,t), ψ(f,f b)} as a collection γ(T,t) : F(T,t) → E(T,t) for each (T, t) such that the diagram

f ∗γ ∗ (T,t) ∗ f F(T,t) f E(T,t)

φ(f,fb) ψ(f,fb) γ(T 0,t0) F(T 0,t0) E(T 0,t0) commutes.

Definition 2.10. Let X be an algebraic stack. A sheaf F of OX-modules on Lis-Et(´ X) is cartesian if for every morphism (f, f b):(T 0, t0) → (T, t) 0 ∗ the map of OT -modules φ(f,f b) : f F(T,t) → F(T 0,t0) is an isomorphism. A sheaf on Lis-Et(´ X) of OX -modules F is quasi-coherent if F is cartesian and for every (T, t) the sheaf F(T,t) is a quasi-coherent sheaf on T . If X is locally Noetherian, then for any (T, t) ∈ Lis-Et(´ X) the scheme T is locally Noetherian, and we define a quasi-coherent F on X to be coherent if each

F(T,t) is coherent.

Proof of 2.1. Let pt = Schk. By Proposition 2.5, it is enough to construct an equivalence

{Quasi-coherent sheaves on BG} ↔ QcohG(pt).

19 Step 1: Let (F, σ) be an object of QcohG(pt), then we will construct a quasi-coherent sheaf F on BG by giving data {F(T,t), φ(f,f b)} (Remark 2.9). Step 1.1: Let (T, t) be an object in Lis-Et(´ BG), and P the G-bundle on T defining t: T → BG. We define F(T,t) or F(T,P) as follows. Let f : pt → BG be the morphism defined by the trivial bundle, then we have the pullback square P T

b t f pt BG.

∗ ∗ By Lemma 2.4, the pullback (b F, (idG ×b) σ) is again a G-equivariant quasi- coherent sheaf on P. We will descend it along P → T , see Definition 1.3.

Write a: G ×k P → P for the G-action on G-bundle P, then the map a×pr2 : G×kP → P×kP is an isomorphism. Therefore, the map τ defined as −1 ∗ ∗ ∗ ∗ ∗ ∗ τ := ((a × pr2) ) (idG ×b) σ is an isomorphism pr1b F → pr2b F . The G- equivariance structure on b∗F ensures that τ satisfies the cocycle condition and thus specifies a descent datum (b∗F, τ). The map P → T is an fppf morphism, and so this descent datum is effective, Theorem 4.3.12 in [2], so we obtain a quasi-coherent sheaf F(T,P) on T . Step 1.2: Let (f, f b):(T 0, t0) → (T, t) be a morphism in Lis-Et(´ BG). The morphism (f, f b) is the composition

b (id 0 ,f ) (f,id) (T 0, P0) T→ (T 0, f ∗P) → (T, P),

∗ and so we will construct the isomorphism φ(f,f b) : f F(T,t) → F(T,t) via ∗ φ b ◦ f φ . (idT 0 ,f ) (f,id) Consider (id, f b):(T 0, P0) → (T 0, P). Because G-bundles are locally triv- ial, we consider an ´etalecover U → T 0, where both bundles are trivial. An isomorphism of trivial G-bundles is given by multiplication mg by a certain element g ∈ G(U). (To be precise, mg : G ×k U → G ×k U is the map (a ◦ (idG ×g)) × prU for a the action on G.) In the diagram e mg G ×k U G ×k U U

b pt BG the square is a pullback square and e: U → G ×k U is the section given by ∗ ∗ ∗ ∗ ∗ the unit in G. The sheaves F(U,P) and F(U,P0) equal e b F and e mgb F respectively and we define isomorphism φ(idU ,mg) as ∗ ∗ ∗ ∼ ∗ ∼ ∗ σg ∼ ∗ ∗ e mgb F = (b ◦ mg ◦ e) F = ρgFU → FU = e b F.

20 By descent for an ´etalecover, see e.g. notes [12], this also defines φ b . (idT 0 ,f ) Consider (f, id): (T 0, f ∗P) → (T, P). We have the diagram

f ∗P T 0

b0 f P T

b pt BG where all squares are pullback squares. By definition of F(T 0,f ∗P), its pull- back to f ∗P is defined by (b ◦ b0)∗F =∼ b0∗b∗F , but also the pullback of ∗ ∗ f F(T,P) to f P is by commutativity of the upper square the pullback along 0 ∗ b of b F . Define φ(f,id) to be the natural isomorphism between F(T 0,f ∗P) ∗ and f F(T,P). Step 1.3: Because we have mapped all morphisms to isomorphisms, F is cartesian and, as all F(T,P) are quasi-coherent, this data yields a quasi- coherent sheaf F on BG. Step 2: Let F be a quasi-coherent sheaf on BG that is given by data

{F(T,t), φ(f,f b)}. Let Ge be the trivial G-bundle on pt, and consider the quasi- coherent sheaf F := F on pt. Let p: G → pt be the trivial map. We (pt,Ge) will construct an automorphism σ of p∗F such that (F, σ) ∈ QcohG(pt).

Consider G ×k G as G-bundle on G via pr2 : G ×k G → G. ‘Multiplica- tion by G’ defined as m: G ×k G → G ×k G, (g, h) 7→ (gh, h) is a morphism of G-bundles over G, giving an isomorphism φ(id,m) of F(G,G×kG). Also, b b there is a morphism (p, p ):(G, G ×k G) → (pt, Ge), where p is an isomor- ∗ ∼ ∗ ∗ phism induced by (p ◦ t) = t p , giving an isomorphism φ(p,pb). Define an ∗ isomorphism σ ∈ AutG(p F ) as the composition

−1 φ b φ φ b p∗F −→(p,p ) F −→(id,m) F −→(p,p ) p∗F . (pt,Ge) (G,G×kG) (G,G×kG) (pt,Ge)

0 ∗ 0 For a k-scheme T and g, g ∈ G(T ), we have for σg = g σ that σg ◦σg = σg0g, ∗ because g φ(id,m) is the map induced by mg. Hence, σ gives G-equivariance structure for F .

Step 3: For a morphism γ(T,t) as in Remark 2.9 between quasi-coherent sheaves on BG defined by {F(T,t), φ(f,f b)} and {E(T,t), ψ(f,f b)}, we take the ˜ (pt, G) component to define the morphism γ(pt,G˜) : F(pt,G˜) → E(pt,G˜) of G- equivariant quasi-coherent sheaves. Given a morphism h:(F, σ) → (E, τ) ∗ of G-equivariant quasi-coherent sheaves, we built γ(T,P) by descending b h between quasi-coherent sheaves on P. These constructions give an equivalence.

21 Remark 2.11. Note that under this equivalence, coherent sheaves corre- spond to finite-dimensional vector spaces.

2.3 Quasi-coherent sheaves on BGm

We specifically want to study coherent sheaves on BGm, and so by Theorem 2.1, we want to study representations of the group scheme Gm over a field k. To classify the representations of Gm, we need the following notation. Definition 2.12. Let G = Spec(A) be an affine group scheme over k.A right A-comodule is a k-linear map r : V → V ⊗k A for a k-vector space V such that r r V V ⊗k A V V ⊗k A

r idV ⊗∆ idV ⊗ idV r⊗idA ∼ V ⊗k A V ⊗k A ⊗k A V ⊗k k = V commute, where ∆: A → A ⊗k A is the diagonal map and : A → k is the coidentity. The map r is called the co-action. Let (V, r) be an A-comodule. An A-subcomodule is a k-subspace W ⊂ V such that r(W ) ⊂ W ⊗k A. Then (W, r|W ) is again an A-comodule.

Lemma 2.13. If G = Spec(A) is an affine group scheme over k and V is a finite-dimensional k-vector space, then there is a one to one correspondence between A-comodules (V, r) and representations ρ: G → Autk(V ).

Proof. Let dim V = n, then a choice of basis {e1, ..., en} for V gives an n ∼ isomorphism Autk(V ) → GLn,k and also A = V ⊗k A. Let ρ: G → GLn be a representation of G. Then the map idG gives an a ∈ GLn(k), that is a map

−1 k[Xij, det(Xij) ] → A

n ∼ and so an A-linear endomorphism r(a) of A = V ⊗k A. This r(a) is uniquely determined by its restriction to a k-linear r : V → V ⊗k A, and this gives a A-comodule (V, r). (Because given a restriction r : V → V ⊗k A, we obtain r(a): V ⊗k A → V ⊗k A via r ⊗ idA postcomposed with idV ⊗m, where m: A ⊗k A → A is multiplication.) The conditions of a comodule map follow from the fact that ρ is a group homomorphism. Pn Let r : V → V ⊗k A be a comodule given by ej 7→ i=1 ei ⊗ aij where (aij)1≤i,j≤n ∈ GLn(A) is an invertible n by n matrix. It corresponds to

22 the representation G → GLn sending g ∈ G(T ) sog ¯: A → OT (T ) to (¯g(aij))1≤i,j≤n by which we mean:

ρ⊗idOT idV ⊗g¯⊗idOT idV ⊗m V ⊗k OT −→ V ⊗k A ⊗k OT −→ V ⊗k OT ⊗k OT −→ V ⊗k OT .

For Gm we may classify the representations on vector spaces V by their weight.

−1 Definition 2.14. For m ∈ Z, a k[X,X ]-comodule (V, r) has weight m if it equals −1 m r : V → V ⊗k k[X,X ], v 7→ v ⊗ X . −1 A weight m representation of Gm is a representation induced by a k[X,X ]- comodule of weight m, via Lemma 2.13. The following proposition is 4.4 in [13]. Proposition 2.15. Every k[X,X−1]-comodule (V, r) is a direct sum L V m∈Z m of weight m subcomodules Vm.

Proof. For each m ∈ Z, define m Vm = {v ∈ V : r(v) = v ⊗ X }, which is by definition weight m and a subcomodule. To show V = L V , m∈Z m n −1 let v ∈ V and because {X | n ∈ Z} form a k-basis of k[X,X ], we write −1 r(v) ∈ V ⊗k k[X,X ] as X m r(v) = vm ⊗ X m∈Z for vm ∈ V . Using the properties of r as comodule: we have

(idV ⊗∆) ◦ r(v) = (r ⊗ idA) ◦ r(v), so X m m X m vm ⊗ X ⊗ X = r(vm) ⊗ X . m∈Z m∈Z m Then we have r(vm) = vm ⊗ X and so vm ∈ Vm. Also, we have

(idV ⊗) ◦ r = idV , so X m v = id(v) = (idV ⊗) ◦ r(v) = vm ⊗ (X ), m∈Z m ∼ where (X ) are just scalars in k so under V ⊗k k = V this corresponds to P (Xm)v ∈ L V . Hence we have V = L V . m∈Z m m∈Z m m∈Z m

23 A subrepresentation of a representation ρ: G → Autk(V ), is a linear subspace W ⊂ V such that ρ|W : G → Autk(W ) defined by ρ|W (g) := ρ(g)|W is again a representation.

Corollary 2.16. Every representation ρ: Gm → Autk(V ) is the direct sum of weight m subrepresentations for m ∈ Z. Because all weight m representations are given by diagonal matrices, all finite-dimensional representations of Gm are direct sums of the one- dimensional representations. Also by Proposition 2.15, up to isomorphism, there are Z-many one-dimensional representations of Gm. Hence all finite- dimensional representations of Gm may be expressed as direct sums, tensor products and duals of the weight 1, one-dimensional representation, called the standard representation.

24 3 Moduli stack of Gieseker bundles

3.1 Stable curves A reference for the stack of stable curves is [7] which mainly discusses stable curves of genus zero, and [8] which is the more classic reference for the coarse moduli space Mg,n for stable n-pointed curves of arithmetic genus g.

Definition 3.1. For n ∈ Z≥0, a prestable n-pointed curve (C, p1, ..., pn) over an algebraically closed field k¯ is a connected curve C over k¯ whose only singularities are nodal singularities (i.e., ordinary double points), with a choice of n distinct smooth points p1, ..., pn ∈ C called marks. By a special point we mean a node or a mark. A stable n-pointed curve is a prestable n-pointed curve satisfying the following stability condition: every genus-0 component of the normalisation must have at least 3 special points and every genus-1 component must have at least one special point. Let B be a scheme over a field k.A (pre)stable n-pointed curve over B is a flat and proper map, locally of finite presentation, π : C → B with n disjoint sections, whose geometric fibers π−1(b) are (pre)stable n-pointed curves.

The condition that a genus-1 component must have at least one special point ensures that the space M1,0 of stable curves with zero points and genus 1 is empty. Given an ordered set I, we may also consider I-marked curves with a choice of distinct smooth sections pi ∈ C for i ∈ I.

Example 3.2. Over an algebraically closed field and in genus zero, a stable n-pointed rational curve may be described as follows. A tree of projective lines is a connected curve such that each irreducible component is isomorphic to a projective line, the points of intersection of the components are ordinary double points, and there are no closed circuits: if a node is removed the curve becomes disconnected. The word twig is used for the irreducible components of a tree. A stable n-pointed rational curve then is a tree C of projective lines with n distinct marks, which are smooth points of C, such that every twig has at least three special points.

Definition 3.3. An isomorphism of two n-pointed curves (C, p1, ..., pn) and 0 0 0 ∼ 0 (C , p1, ..., pn) over scheme B is an isomorphism of curves φ: C → C over 0 B such that φ(pi) = pi. Thus an automorphism of (C, p1, ..., pn) is an automorphism of C fixing each mark.

Example 3.4. In genus zero, the stability condition is equivalent to say- ing that there are no non-trivial automorphisms. An automorphism must

25 map a twig with a mark onto itself, and so by induction each twig onto itself. Hence an automorphism is a gluing of automorphisms on each twig and it must fix all marks and nodes. Then the condition that there are three special points on each twig is equivalent to there being no non-trivial automorphisms. Note that stable n-pointed curves of genus g > 0 can have non-trivial automorphisms, but only finitely many. A theorem by Knudsen and Mumford, Theorem 2.7 in [8], is that for each n ≥ 3, there is a smooth projective variety M0,n which is a fine moduli space for stable n-pointed rational curves.

Lemma 3.5. Given a stable n-pointed curve (C, p1, ..., pn) over an alge- braically closed field and an arbitrary point q ∈ C, there is a canonical way to get a stable (n + 1)-pointed curve.

Proof. If q is not a special point, we can simply put pn+1 = q so that (C, p1, ..., pn+1) is a stable (n + 1)-pointed curve. If q is a node, then we can pull the two components apart and add a projective line joining those two points to obtain a new curve and putting the new mark pn+1 anywhere on this line but not on the nodes. If q coincides with one of the marks pi, then we can glue a projective line to the original curve at this point, and place marks pi and pn+1 anywhere on this new line but not on the node. For any two choices, you get a unique isomorphism of the resulting stable (n + 1)-pointed curves. The situations are sketched in Figure 1.

Definition 3.6. This process to obtain a new (n + 1)-pointed curve from the data (C, p1, ..., pn, q) is called stabilisation.

Definition 2.3 in [8] tells us that stabilisation also works for general curves.

Proposition 3.7. Given a stable n-pointed curve (π : C → B, σ1, ..., σn) and let d: B → C be an arbitrary extra section. Then there is a stable 0 0 0 0 0 (n + 1)-pointed curve (C → B, σ1, ..., σn, σn+1) and a morphism φ: C → C −1 ∼ 0 of B-schemes such that φ (C \d) → C \d is an isomorphism, φ◦σn+1 = d, 0 and φ ◦ σi = σi for i = 1, ..., n. Up to isomorphism this curve is unique and is called the stabilisation of (C → B, σ1, ..., σn, δ). Futhermore, stabilisation commutes with fiber products.

Inverse to stabilisation is a process called either contraction or forgetting a section. Given a stable (n + 1)-pointed curve (C, p1, ..., pn, pn+1) over an algebraically closed field, we forget pn+1 and obtain a stable n-pointed curve by firstly removing pn+1 and then contracting any unstable twig if

26 add q and stabilize → q q

→ forget q and contract

add q and stabilize →

pi q q → forget q and contract pi

Figure 1: Stabilisation and contraction in genus 0. such appears. This is illustrated in Figure 1. Forgetting a section again also works for curves over a scheme B. Given a prestable marked curve C, there is a stabilisation morphism st: C → Cst from C to its stabilisation, that blows down every unstable rational curve in C.

3.2 Stack of principal bundles

The reason we do not want to work with the stack BunGm (g, I) of principal Gm-bundles on stable marked curves of type (g, I), is that it does not satisfy the existence part of the valuative criterion for properness, [9, tag 0CLY], this is stated in Remark 1.5 in [6] (where the term valuative criterion for completeness is used). We illustrate this in the example below for g = 0 and #I = 4, using that Gm-bundles are equivalent to line bundles from Proposition 1.29.

2 Example 3.8. Let k be an algebraically closed field. Consider inside Pk 2 the variety Y that on an affine open is given by V (xy) ⊂ U ⊂ Pk, which is a reducible variety, namely two lines meeting at the origin. Let R := k[b] and B = Spec(R), and consider the Σ = B × Y over B by the natural projection 2 morphism. The coordinate ring of Σ on an open inside B × P is given by A := R[x, y]/(xy) and the fiber over each point in b0 ∈ B is isomorphic to the pullback of Y to the residue field of b0. Consider the sheaf of ideals J = (y, x − b). We restrict the ideal sheaf ∼ to the fiber over b0, so considering J ⊗R R/(b − b0) over A ⊗R R/(b − b0) =

27 Σ

1 b = 0 Ab

Figure 2: Sketch of space Σ and points defined by (y, x − b).

k[x, y]/(xy). On each of the fibers outside b0 = 0 this is a smooth point and so it is an invertible sheaf. However, over b0 = 0 the point is not smooth, as illustrated in Figure 2, and the sheaf is actually not invertible.

Consider Σ as a stable marked curve by adding marks σ1, σ2, σ3, σ4, two on each twig. Everywhere outside b = 0, Σ → B together with the line bundle associated J specifies an object in the stack of principal Gm-bundles, but not at b = 0.

3.3 Gieseker bundles To create a space that does satisfy this valuative criterion, we will enlarge the classification problem by looking at Gieseker bundles. Here we use the introduction given in [6]. Firstly, instead of stable curves Σ, unstable twigs may be inserted at the nodes of a stable curve.

Definition 3.9. A morphism m: C → Σ of prestable marked curves over an algebraically closed field k¯ is a modification if m is an isomorphism away from the preimage of the nodes of Σ, and the preimage under m of every 1 node in Σ is either a node or a P with two special points. A modification of a prestable marked curve f :Σ → B is a morphism m: C → Σ such that for each geometric fiber b ∈ B, mb : Cb → Σb is a modification.

An example of a modification is given in Figure 3. Note that at marked points, a modification is an isomorphism and so marked points lift uniquely to the modification. Also note that, although there generally are different modifications C of a stable marked curve Σ, the modification map will always be the stabilisation map.

28 C → Σ

Figure 3: Sketch of a modification in genus 0.

Definition 3.10. Let (Σ, σi) be a stable marked curve over an algebraically closed field k.A Gieseker Gm-bundle on (Σ, σi) is a pair (m, P) consisting of a modification m:(C, σi) → (Σ, σi) and a principal Gm-bundle p: P → C 1 satisfying the Gieseker condition: the restriction of P to an unstable P in C has degree 1. If C → B is a prestable marked curve and Σ → B its stabilisation, and P a bundle on C satisfying the Gieseker condition on all geometric fibers, then (C, σi, P) is a Gieseker bundle on Σ. Considering Gieseker bundles will solve the issue with the existence part of the valuative criterion, as is shown in Proposition 2.14 in [6]. We will illustrate this by showing how Example 3.8 may be extended to a Gieseker bundle over Σ → B. Example 3.11. Continue with the notation as in Example 3.8. The goal is to write down a Gieseker bundle on Σ over B which restricts to the line bundle given by the sheaf of ideals J = (x, y − b) when b 6= 0. The objective is to make a modification C → Σ which leaves all fibers Σb0 intact for b0 6= 0 1 and inserts a P at the node in the fiber Σ0. Inspired by the proof in Knudsen [8] on the existence of stabilisation morphisms, we will explicitly construct the modification as a blow-up. Firstly consider R0 = k[a, b] and A0 = R0[x, y]/(xy − ab) and consider Spec(A0) → Spec(R0). (If a = 0, we have the curve Σ → B.) Consider the sheaf of ideals J = (x−b, y−a), which defines a section Spec(R0) → Spec(A0) by A0 → R0, x 7→ b, y 7→ a. Then we consider the graded A-algebra

0 0 A˜ = A [t1, t2]/(xt2 − bt1, yt1 − at2) ˜0 0 ˜0 where deg ti = 1 and the morphism π : ProjAA → SpecA . Note that as A is finitely generated, the map π is projective. Also the map is birational: on

29 the affine patch t2 6= 0 we have an inverse on function fields by

t1 7→ x/b, t2 7→ yt1/a = (yx)/(ab).

As it is a projective birational map to an affine Noetherian scheme, it actually is a blow-up along a closed subscheme. Also the section defined by the sheaf of ideals lifts by A˜0 → R, x 7→ b, y 7→ a, t 7→ 1 where we denote ti=1 j the patch t 6= 0 by A˜0 and this section passes through the smooth locus i ti=1 of the fiber. Therefore, up to setting a = 0, this is the modification map we are looking for. 0 0 0 To obtain a = 0, we do a base change ×R0 R /a. We have A ×R0 R /a = A and ˜ ˜0 0 A := A ×R0 R /a = k[b, x, y, t1, t2](xy, xt2 − bt1, yt1) and the map p: Proj(A˜) → Spec(A) induced by π. A projective map stays projective under base change, however a birational map need not stay bira- tional. We will show that it is an isomorphism away from (b, x, y). We cover Proj(A˜) by     ∼ k[b, x, y, t2/t1] ∼ k[b, x, t2/t1] D+(t1) = Spec = Spec (x(t2/t1) − b, y, xy) (x(t2/t1) − b) where the A-algebra structure on A˜1 = k[b, x, t2/t1]/(x(t2/t1) − b) is given by A → A˜1, b 7→ b, x 7→ x, y 7→ 0, and     ∼ k[b, x, y, t1/t2] ∼ k[b, y, t1/t2] D+(t2) = Spec = Spec (x − b(t1/t2), y(t1/t2), xy) (y · t1/t2) where the A-algebra structure on A˜2 = k[b, y, t1/t2]/(y · t1/t2) is given by A → A˜2, b 7→ b, x 7→ b · t1/t2, y 7→ y. Now we will check that the fiber of p 1 over the point (x, y, b) is a copy of P . The fiber over the point corresponding to (b, x, y) is, on the affine patch Spec(A˜1), given by the Spec of ∼ A˜1 ⊗A A/(b, x, y) = k[b, x, t2/t1]/(x(t2/t1)−b)×A k[b, x, y]/(b, x, y) = k[t2/t1] and on the patch Spec(A˜2) the fiber is the Spec of ∼ A˜2 ⊗A A/(b, x, y) = k[b, y, t1/t2]/(y(t1/t2)) ×A k[b, x, y]/(b, x, y) = k[t1/t2].

1 Therefore the fiber equals a P . To show that it is an isomorphism away from the preimage of (x, y, b), so where either b 6= 0 or x 6= 0 or y 6= 0, we will show that it is an isomorphism when we can invert one of b, x and y.

30 Firstly, when b 6= 0, the structure map A → A˜2 inducing p will be

−1 −1 k[b, x, y, b ]/(xy) → k[b, y, t1/t2, b ]/(y · t1/t2), where c 7→ b · t1/t2, which is an isomorphism as the inverse is given by −1 −1 −1 b 7→ b, b 7→ b , y 7→ y, t1/t2 7→ xb . Hence this map is open. The structure map A → A˜1 when inverting b is given by

−1 −1 k[b, x, y, b ]/(xy) → k[b, x, t2/t1, b ]/(x · t1/t2 − b) where y 7→ 0. This map is actually inverting x as we have an isomorphism

−1 ∼ −1 −1 k[b, x, t2/t1, b ]/(x · t1/t2 − b) → k[b, x, y, b , x ]/(xy) by t 7→ bx−1 and 0 7→ y and tb−1 7→ x−1. Therefore, the image of the map

Spec(A˜1) → Spec(A) is D(x) and so it is an open map. Also the two maps are jointly surjective, and both open, so the map p where b 6= 0 is ´etale. Then it is also finite locally free and flat, so the degree is locally constant and by connectedness there is a single degree. By computing a fiber, the degree is 1 and so it is an isomorphism where b 6= 0. We can similarly check this when x 6= 0 and y 6= 0. Therefore, p: Proj(A˜) → Spec(A) is a modification of Σ over B. Let C := Proj(A˜). We give a line bundle by taking O(−D) for D = (x − b, y). The section defined by this sheaf of ideals can be extended to one through the smooth locus of C. Because on each fiber it is the line bundle associated to a point, i.e., a divisor of degree 1, the degree of the bundle, also at the 1 unstable P is 1, so the Gieseker condition is satisfied. Hence this specifies the Gieseker bundles solving the existence part of the valuative criterion for Example 3.8.

3.4 Stack of Gieseker Gm-bundles We can define the stack of Gieseker bundles, as given in [6].

Definition 3.12. The stack Me g,I (BGm) of Gieseker Gm-bundles on stable genus g, I-marked curves is the fibered category over k-schemes with objects

(B, π : C → B, σi, P) where • B is a k-scheme, and

• π : C → B is a pre-stable genus g curve with marked points σi : B → C, and

•P is a principal Gm-bundle p: P → C defining a Gieseker bundle on the stabilisation C → Cst.

31 Morphisms in the category are triples (b: B0 → B, f : C0 → C, f˜: P0 → P) 0 0 such that C = B ×B C, the diagram

f˜ P0 P p0 p f C0 C 0 σi σi π0 π B0 b B ˜ commutes, and f is Gm-equivariant. In the article [6] Corollary 2.10, it is proven that it is in fact an Artin stack. Also, some basic facts about the geometry of the stack are mentioned, such as that it is of dimension (g − 1) + 3(g − 1) + #I. Before we carefully discuss Me g,I (BGm) in genus 0 with #I = 3 marks, we need the following lemma. Consider the k-scheme T and the section 1 σ∞ : T → PT ; the image of the section is an effective relative Cartier divisor 1 of degree 1 on PT . As σ∞ defines a smooth point in each fiber, the sheaf O 1 (−σ∞) defined by the ideal sheaf given by σ∞, is actually a line bundle. PT ⊗−D 1 We may also consider O(Dσ∞) = O(−σ∞) on PT , and these sheaves satisfy the following.

0 1 Lemma 3.13. Let f : T → T be a morphism of k-schemes, σ∞ : T → PT 0 0 1 1 ¯ 1 1 and σ∞ : T → PT 0 sections induced by the point ∞ in Pk, and f : PT 0 → PT ¯∗ 0 the projection map. Then we have f O(Dσ∞) = O(Dσ∞) as submodules of ∗ O 1 = f¯ O 1 . PT 0 PT ¯∗ 0 Proof. It is enough to show that f O(−σ∞) = O(−σ∞). Reducing to the affine case, let g : X = Spec(B) → Y = Spec(A) and Cartier divisors σX of X and σY of Y be given, such that

σX σY

g X Y is a pullback diagram. Let f op : A → B, and I the ideal of A corresponding to σY , then the ideal of B corresponding to σX is IB because the diagram

A B

A/I B/IB

32 ∗ is a pushout diagram. Then g I = I ⊗A B and we have natural surjection I ⊗A B → IB. Both ideals yield an invertible sheaf, that is a line bundle, ∼ and a surjection of line bundles is an isomorphism, so we have I ⊗A B = IB ∗ and g O(−σY ) = O(−σX ).

Remark 3.14. The proof in Lemma 3.13 actually proves something more general: given a closed subscheme locally cut out by a regular element, and its pullback is also locally cut out by a regular element, then the pullback of the ideal sheaf is the ideal sheaf of the pullback.

Example 3.15. The easiest example of the stack Me g,I (BGm) is where #I = 3 and the curves are genus 0, so Me 0,3(BGm). There is, up to unique isomorphism, only one stable genus zero curve with 3 marked points over k, 1 namely simply (Pk, 0, 1, ∞), because we have only three marks to distribute. Because this curve has no nodes, the only modifications of this curve are 1 1 automorphisms of Pk. Because this automorphism of Pk has to preserve the three marked points, the only possible modification is the trivial modifica- tion m = id. We have also established in Example 1.28 that there is only one 1 line bundle of set degree D on , namely O 1 (D). Hence we may describe Pk Pk the stack explicitly in Proposition 3.18.

Definition 3.16. Define a functor G Ψ: Me 0,3(BGm) → BGm D∈Z as follows. Write i : B → F B for the inclusion for the degree D Gm D∈Z Gm ∼ 1 D component. Given an object (T,CT = PT , σi, P) ∈ Me 0,3(BGm)(T ) with P of degree D, then we define

∗ (T,CT , σi, P) 7→ iD(T, σ1P).

If P is not of one degree, then the degree of P is a locally constant function d: T → and T = F d−1(a). Let T := d−1(a), then by the universal Z a∈Z a property of the disjoint union, a map from T = taTa is defined by maps from each Ta. Then we map (T,CT , σi, P) to the element associated to the collection i (T , σ |∗ P| ) for a ∈ . a a 1 Ta Ta Z 0 0 0 A morphism φ:(T ,CT 0 , σi, P ) → (T,CT , σi, P) as defined in 3.12 where 0 0 CT 0 = CT ×T T and P, P are assumed to have degree D, specifies an iso- b 0 ∗ 0 0∗ 0 morphism f : P → f P. Then we map φ to the morphism from (T , σ1 P ) ∗ 0∗ b to (T, σ1P) given by the pair (t, σ1 f ), because we have

0∗ b 0∗ 0 0∗ ∗ ∼ 0 ∗ ∼ ∗ ∼ ∗ ∗ σ1 f : σ1 P → σ1 f P = (f ◦ σi) P = (σ1 ◦ t) = t σ1P.

33 Definition 3.17. Define a functor G Ξ: BGm → Me 0,3(BGm) D∈Z by defining Ξ ◦ iD for any D. On objects we define

1 ∗ (T, L) 7→ (T, π : → T, σ0, σ1, σ∞, P := O(Dσ∞) ⊗ π L). T PT L OCT T

For a morphism (f, f b):(T 0, L0) → (T, L), so f : T 0 → T and f b : f ∗L →∼ L0, we define an isomorphism of line bundles

¯∗  ∗  ∼ 0 ∗ 0 f O(Dσ∞) ⊗ π L → O(Dσ ) ⊗ π 0 L OCT T ∞ OCT T

¯∗ 0 b by f O(Dσ∞) = O(Dσ∞) from Lemma 3.13 and f . Proposition 3.18. The functor Ψ gives an isomorphism of stacks ∼ G Me 0,3(BGm) = BGm. D∈Z Proof. We show that Ψ and Ξ are inverses. Let L be any line bundle on ∗ T , then σ1PL is naturally isomorphic to L, because σ1 and σ∞ are disjoint sections and so

∗ ∗ ∗ ∗ ∼ ∼ σ1PL = σ1O(Dσ∞) ⊗OT σ1πT L = OT ⊗OT L = L.

Secondly, consider an object (T,CT , σi, P) in Me 0,3(BGm) where P is of ∼ 1 degree D ∈ Z and (CT , σi) = (PT , σ0, σ1, σ∞). We want to prove that there is a natural isomorphism

∼ ∗ ∗ P = O(Dσ∞) ⊗ π σ P. OCT T 1 For this, we consider the sheaf

∗ ∗ ∨ F = O(Dσ∞) ⊗ π σ P ⊗ P OCT T 1 OCT

1 on CT = and we will show it is O 1 . Note that F trivialises along σ1 and PT PT so it is enough to trivialise locally on T. Therefore, without loss of generality, assume T is local. On the fiber over the closed point of T , F is trivial, 1 because it has degree 0 on Pk. This trivialisation is given by some global section α. We use [5] Theorem III.12.11 (Cohomology and Base Change) 1 1 for πT : PT → T and F on PT and y the closed point of T . Because Fy is 1 1 0 1 trivial, we know that dim H ((PT )y,Fy) = 0 and dim H ((PT )y,Fy) = 1. Part (a) of the theorem for i = 1 implies that for the fibre we have 1 1 R πT ∗(F ) ⊗ k(y) = 0, as φ is clearly surjective and so an isomorphism.

34 1 Then also R πT ∗(F ) = 0 by Nakayama’s lemma and finite generation of cohomology (which follows from properness of πT ). Then part (b) of the theorem implies that φ0 is surjective, and we may 0 0 0 choose an α ∈ R πT ∗(F ) ⊗ k(y) mapping to α. Then α gives a global section and so defines a map O 1 → F , which restricts to alpha over the PT closed point. Over the closed point, the map is surjective, and then by

Nakayama’s lemma the map O 1 → F is a surjective map of line bundles. PT Because a surjective map of line bundles is an isomorphism, we indeed have ∼ ∼ ∗ ∗ F = O 1 and P = O(Dσ∞) ⊗O π σ P. PT CT T 1 Then we indeed have constructed an explicit inverse, and we conclude

∼ G Me 0,3(BGm) = BGm. D∈Z

35 4 Gromov-Witten invariants

4.1 Evaluation maps

In this section, we define the evaluation maps on the stack Me := Me g,I (BGm), and equip Me with a natural curve, a line bundle, and sections. That is, we construct the following diagram and examine them for Me 0,3(BGm).

P

φ C BGm σi evi π (2)

Me g,I (BGm)

F

Mg,I

These constructions are standard, and if a reader is familiar with these and does not wish to see the details, he or she may continue reading at section 4.2. If Me were a scheme, then we could by Yoneda’s lemma associate to id ∈ Hom(M, M) an element in M(M) and so a triple Me e e e e (C , σ , P), Me i which would be the natural curve, sections and line bundle. In this case we have the following stack as natural curve.

Definition 4.1. Let Me := Me g,I (BGm), and define C → Me as the fibered category given by

C(T ) = {(CT → T, σi, P, c) | (CT , σi, P) ∈ Me (T ), c ∈ CT (T )} for T a k-scheme. This means we consider Me with an extra section which need not be disjoint from the σi. Morphisms are commutative diagrams like morphisms in Me , see Definition 3.12, where also the final section commutes. The map C → Me is the natural forgetful morphism

π : C → Me , (T,CT , σi, P, c) 7→ (T,CT , σi, P) forgetting the final section.

Lemma 4.2. The map π : C → Me is representable.

36 Proof. Let T → Me be given, specified by (CT , σi, P), and consider any map f : T 0 → T . By properties of the fiber product we have 0 ∼ 0 CT 0 (T ) = CT (T ), and C × T 0(T 0) =∼ C × T × T 0(T 0) =∼ C × T (T 0). Me Me T Me 0 0 ∗ The composition T → T → Me is given by (CT 0 = CT ×T T , σi ◦ f, f P), and then the definition of C gives that

0 0 0 ∼ ∗ C × T (T ) = {x ∈ C(T ), π(x) → (C 0 , σ ◦ f, f P)} Me T i and so we only add a section of CT 0 . We conclude

0 0 0 0 0 C × T (T ) =∼ C × T (T ) =∼ C 0 (T ) =∼ C (T ). Me Me T T Hence π : C → Me is representable by schemes.

This also explains why we would call this a natural ‘curve’ over Me ; for all T → M the fiber product C × T is the curve C over T . e Me T We also define for j ∈ I the section maps σj to C by

σj : Me → C, (T,CT , σi, P) 7→ (T,CT , σi, P, σj). Moreover, we have a natural classifying map

φ: C → BGm ∗ sending an object (T,CT , σi, P, c) to (T, c P). Suppose we have a morphism 0 0 0 0 0 from (T ,CT 0 = CT ×T T , σi, P , c ) to (T,CT , σi, P, c), so a commutative 0 ¯ ˜ ¯∗ 0 diagram as in 3.12 with f : T → T, f : CT 0 → CT and f : f P → P . Then we map it to (f, c0∗f˜):(T 0, c0∗P0) → (T, c∗P) where c∗f˜ is indeed an isomor- phism of c0∗f¯∗P =∼ f ∗c∗P and c0∗P0 as f˜◦ c0 = c ◦ f. This map specifies a line bundle P on C given as sheaf on the stack C by sending an object (T, t) in Lis-Et(´ C) to the OT -module and line bundle ∗ c P, when t: T → C is specified by (CT , σi, P, c). By the natural triple for Me we then mean (C, σi, P). Now define the evaluation maps evi = φ ◦ σi

evi : Me g,I (BGm) → BGm ∗ by sending (T,CT , σi, P) to (T, σi P). Finally, important to note is the forgetful map

F : Me g,I (BGm) → Mg,I to the stack of stable genus g, I-marked curves, forgetting P and sending

(C, σi) to its stabilisation. Thus we have defined diagram (2).

37 Lemma 4.3. The curve C over Me 0,3(BGm) is given by

1 1 := Me 0,3(B m) × . PMe G M0,3 PM0,3

1 Proof. Firstly, we prove that the universal curve over M0,3 is Pk. We have ∼ the isomorphism M0,3 → Schk, (T,CT , σi) 7→ T with an inverse given by 1 ¯ T 7→ (T, PT , σ0, σ1, σ∞). The curve C over M0,3 on k-scheme T is given by

C¯(T ) = {(CT , σi, c ∈ CT (T )} and C¯ → M0,3 is given by forgetting the last section. The diagram

∼ M0,3 k

¯ 1 C Pk ¯ 1 may be completed with the map C → Pk sending (CT , σi, c) to

c ∼ 1 1 T → CT → PT → Pk.

1 1 1 For the fiber product PT = Pk ×k T , giving s: T → Pk is equivalent to giving 1 1 ¯ a map t: T → PT , so an inverse Pk → C is given by

1 s 7→ (PT , σ0, σ1, σ∞, t).

¯ ∼ 1 This isomorphism C → Pk makes the above diagram commute. Note that ¯ ∼ 1 in this case, one can see that the universal curve C = Pk and its sections is indeed the element of M (k) corresponding to id . 0,3 M0,3 The conclusion for C over Me 0,3(BGm) follows from the pullback square

Fc ¯ ∼ 1 C C = Pk

F Me 0,3(BGm) M0,3 where both F and Fc are forgetting the bundle.

Remark 4.4. We can restrict the (C, σi, P) to degree D components of M (B ) ∼ F B . We have inclusion i : B → F B and e 0,3 Gm = D∈Z Gm D Gm D∈Z Gm by postcomposing with the isomorphism to Me 0,3(BGm) we get

jD : BGm → Me 0,3(BGm),

1 ∗ (T, L) 7→ (T, πT : PT → T, σ0, σ1, σ∞, O(Dσ∞) ⊗O 1 πT L). PT

38 Then we have B × C =∼ 1 Gm Me PBGm with objects triples (T, L, t) of a k-scheme T , a line bundle L on T and an 1 1 extra map s: T → Pk (or t: T → PT ). Then we have the following maps

P BGm φ

ι 1 D C 1 PBGm Pk σi π

σ0 0 F i π Me 0,3(BGm) M0,3 j D ∼ i B D F B Gm D∈Z Gm where π0 is the map induced by the pullback and σ0 : B → 1 are i Gm PBGm 1 given by (T, L) 7→ (T, L, σi : T → PT ) for i = 0, 1, ∞. We can also restrict the natural bundle P to 1 . Note that the map PBGm ιD is defined by

1 1 ∗ 1 (T, L, t: T → PT ) 7→ (T, PT , σ0, σ1, σ∞, O(Dσ∞) ⊗O 1 πT L, t: T → PT ). PT Then we have on objects that

1 ∗ ∗ P(ιD((T, L, t: T → PT ))) = t (O(Dσ∞) ⊗O 1 πT L) PT

∗ and as ιD is an open immersion, this defines ιDP. Note that an evaluation map evi = φ ◦ σi restricted to a degree D com- ponent is actually simply the identity map, because we have

1 ∗ evi ◦jD(T, L) = evi(T, PT , σ0, σ1, σ∞, O(Dσ∞) ⊗O 1 πT L) PT 1 ∗ T 1 = φ(T, PT , σ0, σ1, σ∞, O(Dσ∞) ⊗O 1 πT L, σi : T → PT ) PT T ∗ ∗ = (T, σi (O(Dσ∞) ⊗O 1 πT L)). PT

T T ∗ ∗ ∼ Note that πT ◦ σi = idT and so canonically σi πT L = L. Lemma 3.13 gives for

1 pr 1 PT Pk T π π k σi T k σi T t k

39 ∗ k T that pr O(Dσ∞) = O(Dσ∞), and so T ∗ T ∼ T ∗ ∗ k ∼ ∗ k∗ k ∼ σi O(Dσ∞) = σi pr O(Dσ∞) = t σi O(Dσ∞) = OT ,

k∗ k because σi O(Dσ∞) is a line bundle on k and so is trivial. Hence we have T ∗ ∗ ∼ ∼ evi ◦jD(T, L) = (T, σi (O(Dσ∞) ⊗O 1 πT L)) = (T, OT ⊗OT L) = (T, L). PT

Similarly the map is trivial on morphisms, and we have evi ◦jD = idBGm .

4.2 Gromov-Witten invariants Gromov-Witten invariants can be computed using cohomology classes and the Chow group, for example in the article [7], where the stack of stable genus g, I-marked curves and the stack of stable maps is studied. However, as is motivated in [6], because Me g,I (BGm) is an Artin stack rather than a Deligne-Mumford stack, it is not likely we may integrate cohomology classes along the forgetful morphism F . Instead of cohomology classes, we will push forward K-theory classes of coherent sheaves, which are defined as follows.

Definition 4.5. For an algebraic stack X, the category Coh(X) of coherent sheaves of OX -modules is a full additive subcategory of an abelian category OX − Mod. For C := Coh(X), consider F (C) the free abelian group on the objects of C up to isomorphism. Let (E) : 0 → M 0 → M → M 00 → 0 be an exact sequence in C, and write Q(E) = [M]−[M 00]−[M 0] ∈ F (C). Let H(C) be the subgroup of F (C) generated by the Q(E) for all short exact sequences

E. Then the Grothendieck group K0(X) of Coh(X) is K0(X) = F (C)/H(C).

On Me g,I (BGm), we have the following (classes of) coherent sheaves that are of interest. For any finite-dimensional representation Gm → Autk(V ) of ∗ Gm, we have the coherent sheaf φ V on C. In particular, for the standard ∗ representation Gm1 given by id: Gm → Gm, we have φ (Gm1) is the line bundle attached to the torsor P.

Definition 4.6. For a finite-dimensional representation V of Gm, we call

∗ ∗ ∗ evi [V ] = σi φ V an evaluation bundle on Me g,I (BGm). Its descendant bundles are the classes ∗ ⊗ji ∗ ∨ ev [V ] ⊗ [T ], where Ti = σ Ω is the relative tangent line to C at σi i i i C/Me and ji is an integer.

Remark 4.7. Note that every coherent sheaf on BGm is locally free as this ∗ may be checked on the smooth cover by a point. Thus each evi [Va] is locally

40 ∗ free and so flat. Then, we may take the product of classes evi [V ] to be the tensor product (in that case the quotiented exact sequences are preserved by tensoring).

To define what we, in this thesis, mean by a Gromov-Witten invariant, we firstly define the K-theoretic pushforward.

Definition 4.8. For a morphism f : X → Y , the K-theoretic pushforward of a class of a coherent sheaf F on X is defined as

X i i f∗[F] = (−1) [R f∗F], i if this exists in K0(Y ).

Remark 4.9. Two problems may occur for the existence of this pushforward i in K0(Y ). Firstly, R f∗F may not be coherent, although, if f is proper, it is. Secondly, the sum may have infinitely many non-zero terms.

Definition 4.10. By a Gromov-Witten invariant for Me g,I (BGm) we mean the following. Let F : Me g,I (BGm) → Mg,I be as above and p: Mg,I → pt. A Gromov-Witten invariant is, if it is defined, the K-theoretic pushforward p∗F∗ of the tensor product of a number of evaluation bundles, descendant bundles, an admissible line bundle L, as will be defined in Definition 4.19, i ∗ and complexes R π∗φ V as will be defined in Construction 4.13. A Gromov-Witten invariant then is a class in K0(pt). Note that K0(pt) is naturally isomorphic to Z by taking the dimension of a k-vector space. Thus, we can associate to a Gromov-Witten invariant a number, which we refer to as a numerical Gromov-Witten invariant.

The map p: Mg,I → pt is proper, so the existence of this pushforward is less technical. However, the existence of a pushforward along F to Mg,I , is not straightforward as Me g,I (BGm) is not proper, see Section 2.3 Limits of Bundles in [6].

Example 4.11. For #I = 3 and Me 0,3(BGm), the map ∼ G F : Me 0,3(BGm) = BGm → M0,3 D∈Z is not proper, because we have infinitely many copies of BGm and so the map is not quasi-compact. Since M 0,3 is a point, there is no need to push forward along p.

41 The main theorem in [6] states that for certain specified admissible com- plexes α of coherent sheaves, the derived pushforward RF∗α is a bounded complex of coherent sheaves. Then the K-theoretic pushforward is actually defined, and thus we have Gromov-Witten invariants. In the next sections, we shall try to illustrate this main theorem in the special case of genus 0 curves with 3 marks.

4.3 Admissible line bundles

∗ We want to push forward bundles φ V on C along the map π : C → Me 0,3(BGm) ∗ so in the complex Rπ∗φ V . Basics of the higher direct image or derived pushforward for maps of schemes may be found in Liu, [3], Chapter 5.2.3. We mainly use the following property, Proposition 2.28 in [3]. Lemma 4.12. Let π : X → Spec(k) be a map of schemes, and let L be any i i ∼ line bundle on X, then R π∗(L) is naturally isomorphic to H (X, L) , the i Ok-module associated to H (X, L).

Note that π : C → Me 0,3(BGm) is a map of stacks, not schemes. There- ∗ fore, we will explicitly construct the sheaf Rπ∗φ V . Because Me 0,3(BGm) ∗ is isomorphic to a disjoint union, we can define the sheaf Rπ∗φ V by the restriction to its components, as in Remark 4.4. Construction 4.13. Write F for the coherent sheaf ι∗ φ∗V on 1 . We D PBGm Gm i 0 shall define (E, σ) ∈ Coh (pt) giving a coherent sheaf R π∗F on BGm via Theorem 2.1 and its proof. Using the cover map pt → BGm defined by the trivial line bundle, we have the pullback diagram

1 π˜ Pk pt f (3)

0 1 π B . PBGm Gm 1 where the map f is given by (s: T → Pk) 7→ (T, OT , s). We consider the coherent sheaf on pt given by

i ∗ i 1 ∗ E := R π˜∗f F = H (Pk, f F), via Lemma 4.12. To define the automorphism σ : p∗E → p∗E where p is the map p: Gm → pt, consider the pullback square 1 q 1 Pk ×k Gm Pk

π2 π˜ p Gm pt .

42 Becauseπ ˜ is flat, flat base change gives a canonical isomorphism

∗ ∼ i ∗ ∗ p E = R π2∗q (f F), so we can compare their automorphisms. An automorphism of q∗(f ∗F) will i ∗ ∗ ∗ ∗ induce an automorphism of R π2∗q (f F), and an automorphism of q (f F) 1 1 can be given by an element of Gm(P × Gm). Naively, let α: P × Gm → Gm 1 ∗ be given, then for each a ∈ P ×Gm(k), we get an α(a) ∈ Gm(k) = k . Then a∗(q∗(f ∗F)) is a k-vector space and we get an automorphism by multiplying by α(a). Hence, the automorphism σ of p∗E can be given by an element of 1 Gm(P × Gm).

∗ ∗ ∗ 1 Example 4.14. Consider F = ιDφ Gm1 = ιDP. For s: T → Pk, and 1 1 t: T → PT = Pk ×k T such that pr ◦ t = s, we have

1 ∗  T ∗  ∗ T ∗ ∗ F(f(s: T → Pk)) = t O(Dσ∞) ⊗O 1 πT OT = t (O(Dσ∞)) ⊗OT t πT OT . PT ∼ ∗ T ∼ ∗ T = t (O(Dσ∞)) ⊗OT OT = t (O(Dσ∞)). 1 1 T ∼ ∗ k Then Lemma 3.13 gives for pr: PT → Pk that O(Dσ∞) = pr O(Dσ∞), and so naturally ∗ T ∼ ∗ ∗ k ∼ ∗ k t O(Dσ∞) = t pr O(Dσ∞) = s O(Dσ∞). ∗ Then the sheaf f F on Schk is canonically isomorphic to the line bundle k 1 O(Dσ∞) on Pk, which is isomorphic to the line bundle O(D) from Example 1.28, where D is the degree. Via Riemann-Roch or directly as in Example 1.28 for H0, we compute

 0 if i < 0    D + 1 if D > −2, i = 0  i 1  0 if D < 0, i = 0 dim H (Pk, O(D)) = (4)  0 if D > −2, i = 1   −(D + 1) if D < 0, i = 1   0 if i > 1

i 1 ∗ i ∗ which equals dim H (Pk, f F) = dim E = dim R π˜∗f F. The structure on 1 E of a Gm-equivariant sheaf will be induced by the map π2 : P ×Gm → Gm. ∗ Let g ∈ Gm(k), then we check what the induced automorphism σ of p E does by studying the following diagram, where the dashed arrows indicate bundles.

43 O(D) q∗O(D) O(D)

1 id ×g 1 q 1 Pk Pk ×k Gm Pk

t π2 π˜ g p pt Gm pt

i 1 ∗ i 1 i 1 H (P , O(D)) p H (P , O(D)) H (P , O(D))

Studying the automorphism of q∗O(D), the construction described the ∗ ∗ 1 automorphism as multiplying (id ×g) q O(D) = O(D) on Pk by the unit given by π2(id ×g) which equals g ◦ t and so it is ‘multiplication by g’. 1 The pushforward along t of this multiplication by g on O(D) on Pk, to ∗ ∗ i 1 ∼ i 1 g p H (Pk, O(D)) = H (Pk, O(D)) on pt is again multiplication by g. We will deduce the representation associated to the Gm-equivariant co- i 1 herent sheaf E = H (Pk, O(D)) on pt. The representation namely sends ∗ g ∈ Gm(k) to the automorphism simply multiplying by g in Aut(g E). Thus the representation of Gm, classified in Section 2.3, is simply the weight one i 1 representation of a dimension dim H (P , O(D)) vector space over k. This i 1 is the direct sum of dim H (P , O(D)) copies of the standard representation.

Remark 4.15. Given a general finite-dimensional representation V of Gm, L ⊗nj then V equals j∈J Gm1 for a certain index set J and nj ∈ Z. Now the above construction and example also tell us what representations belong to i ∗ R π˜∗φ V . We have

⊗n ∗ ∗ M ∗ ∗ j ∼ M ∗ ⊗nj ιDφ V = ιDφ Gm1 = ιDP j∈J j∈J on 1 . For f : pt → B defined by the trivial line bundle, f ∗ι∗ φ∗V is PBGm Gm D isomorphic to M O(D)⊗nj j∈J 1 L on Pk. Via Example 1.28, this is isomorphic to j∈J O(nj · D). We defined the sheaf E on pt as

i 1 M ∼ M i 1 H (Pk, O(nj · D)) = H (Pk, O(nj · D)) j∈J j∈J which all vanish when i 6= 0, 1. We want to find the Gm-equivariance action i 1 on each H (Pk, O(nj · D)). Similarly to Example 4.14, but now for a nj- fold tensor product of the standard O(D), we get weight nj representations.

44 i ∗ Thus we get that on the degree D component, R π˜∗φ V is given by the direct i 1 sum over j ∈ J of dimension dim H (P , O(nj ·D)) weight nj representations. In order to define admissible line bundles, we need the concept of the determinant; this definition may be found in Stacks Project [9], tag 0B37, or in [10].

Definition 4.16. Let (X, OX ) be a locally ringed space and E a finite locally free OX -module. Because the rank is locally constant, we obtain a decomposition X = F X where the X are open and closed, such that n∈Z n n E is finite locally free of rank n on Xn. We define the determinant det(E) Vn as the invertible sheaf on X which is equal to the exterior power (E|Xn ) on Xn for all n ≥ 0. Because det(E) is finite locally free of rank 1, it is a line bundle and so det(E) ∈ Pic(X).

Definition 4.17. Let f : X → Y be a proper morphism of schemes over S. For every coherent sheaf F on X that is flat over S, we have the invertible

OY -module det Rf∗F defined as follows. If a finite complex of locally free OY -modules of finite rank E∗ = (0 → E0 → E1 → ... → En → 0) exists such that there is a ∗ quasi-isomorphism E → Rf∗F of complexes, then we have

n O i ⊗(−1)i det Rf∗F = (det E ) . i=0

i When in particular all OY -modules R f∗F for i ≥ 0 are locally free of finite rank, then we have i O i
⊗(−1) det R π∗L . (5) i

The determinant of the zero sheaf is canonically isomorphic to OY .

Remark 4.18. For the proper morphism of stacks π : C → Me , we have i ∗ i defined the coherent sheaf R π∗φ V via R π˜∗φV in Construction 4.13. We define det Rπ φ∗V on each D-th component of M ∼ F B using (5). ∗ e = D∈Z Gm

Definition 4.19. Write Gm1 for the standard representation of Gm. The admissible line bundles L are the non-zero powers of the determinant

∗ ⊗−n L = (det Rπ∗φ Gm1) , for n ∈ Z \{0}.

∗ Example 4.20. We compute det Rπ∗φ Gm1 for Me 0,3(BGm). For degree D and π0 : 1 → B , because all Riπ0 ι∗ P are coherent as ι∗ P is coherent PBGm Gm ∗ D D

45 and π0 is proper (becauseπ ˜ is proper in diagram (3)), we may compute the determinant via i O i 0 ∗
⊗(−1) det R π ∗ιDP . i i 0 ∗ The R π ∗ιDP vanish for i 6= 0, 1, and so we have

0 ∗ 0 1 1 1 ⊗−1 det Rπ ∗ιDP = det(H (P , O(D))) ⊗ det(H (P , O(D)))   OBGm if D = −1  0 1 = det(H (P , O(D))) if D ≥ 0  1 1 ⊗−1  det(H (P , O(D))) if D ≤ −2

0 1 1 1 ∨ As the H (P , O(D)) and H (P , O(D)) are of dimension (D + 1) respec- tively −(D +1) over k, we get a (D +1)-th respectively −(D +1)-th exterior power as determinant. Taking this exterior power, and the dual if D ≤ −2, gives a weight (D + 1) one-dimensional representation of Gm. From this example and the fact that tensoring representations means adding weights, we get the following result.

∗ ⊗−n Lemma 4.21. Let n ∈ Z \{0} and consider L = (det Rπ∗φ Gm1) on ∼ Me 0,3(BGm). Then on the degree D component of Me 0,3(BGm) = tD∈ZBGm, L is given by a one-dimensional weight n(D + 1) representation.

4.4 Coherent pushforward of an admissible complex

Definition 4.22. An admissible complex α on Me g,I (BGm) is the tensor product of an admissible line bundle L with any finite number of Dolbeault indexes and evaluation or descendant bundles:

O ∗ O ∗ ⊗ni
α = L (⊗a∈ARπ∗φ Va) ⊗b∈B evi [Wb] ⊗ [Ti ] for A, B finite sets of indices, ni integers, and Va,Wb finite-dimensional rep- resentations of Gm. We prove the following special case of the Main Theorem 6.1 in [6].

Theorem 4.23. The derived pushforward RF∗α of any admissible complex ∼ α on Me 0,3(BGm) along F : Me 0,3(BGm) → M0,3 = pt is coherent. The strategy to prove the theorem is to show that admissible complexes are bounded in the sense that the pushforward gives the zero class on in- finitely many components of F B ∼ M (B ). For just the D-th D∈Z Gm = e 0,3 Gm component, so for FD := F ◦ jD : BGm → Me 0,3(BGm) → pt, we will show that the pushforward does preserve coherence.

46 Remark 4.24. Note that, using the definitions as in the Stacks Project

[9], this is not because FD is proper [9, tag 0CL5]: FD is not seperated as defined in [9, tag 04YV] because the map Gm → pt and so also ∆FD is not proper.

However, FD∗ does preserve quasi-coherent sheaves, by [9, tag 0704]: FD is a quasi-compact and quasi-separated morphism of algebraic stacks by the following. Firstly Gm → pt is affine and so quasi-seperated and quasi-compact, and so ∆FD is, and so FD is quasi-seperated. Recall that BGm has the smooth surjective map pt → BGm where pt is quasi-compact. Also, BGm is quasi-compact [9, tag 04CC] and pt is quasi-seperated, and so by Lemma 93.7.8(1) [9, tag 050S], FD is quasi-compact. Then FD∗ is via Theorem 2.1 a functor

Repr ( ) → Qcoh(pt). k Gm

We will compare FD to the invariants functor, defined as follows. Definition 4.25. The invariants functor on Repr ( ) is given by the k Gm sheafification of the presheaf

(−)Gm : Repr ( ) → Qcoh(pt) k Gm (V, ρ) 7→ {(t: T → k) 7→ (t∗V )Gm(T )}. Note that if k is an algebraically closed field, then the presheaf is a quasi- coherent sheaf, because (tV )Gm(T ) = t∗(V Gm(k)) holds because of the fol- lowing. We reduce this to weight m representations and if the weight is 0, then (tV )Gm(T ) = t∗V = t∗(V Gm(k)) as everything remains invariant. If the weight in non-zero, the only 0 remains invariant under multiplication by all units and so (tV )Gm(T ) = 0 = t∗0 = t∗(V Gm(k)).

Gm ∗ Proposition 4.26. The functor (−) is right adjoint to FD, and therefore Gm (−) = FD∗. Proof. Firstly F ∗ : Qcoh(pt) → Repr ( ) is the trivial functor, sending D k Gm op F : Schk → O −Mod to the vector space V = F (idk) with the trivial weight 0 representation, so ρtriv : Gm → Autk(W ), g 7→ id. We will show that for all k-vector spaces V and representations (W, ρ: Gm → Autk(W )) of Gm that

∗ ∼ Gm HomRepr (F V, (W, ρ)) = Hom (V, (W, ρ) ). k D Qcoh(pt) On the right-hand side we have a natural transformation and so for each ∗ ∗ m(T ) k-scheme t: T → k a morphism φT : t V → (t W )G . On the left-hand side we have k-linear maps h: V → W such that for all g ∈ Gm(T ) the diagram

47 ρ t∗V triv t∗V

t∗h t∗h ρ(g) t∗W t∗W ∗ ∗ commutes. Therefore, it satisfies ρ(g) ◦ (t h) = t h for all g ∈ Gm(T ), meaning that t∗h should map into (t∗W )Gm(T ). Gm Given φ = {φt: T →k} ∈ HomQcoh(pt)(V, (W, ρ) ), we can send it to ∗ φid ∈ HomRepr (F V, (W, ρ)). The inverse is given by sending a morphism k k D ∗ ∗ m (h: V → W ) ∈ HomRepr (F V, (W, ρ)) to {t h}t: T →k. Thus (−)G is right k D ∗ adjoint to FD. Adjoints, if they exist, are unique up to isomorphism, and ∗ Gm ∼ FD∗ is right adjoint to FD, so we conclude (−) = FD∗.

As (−)Gm sends finite-dimensional representations to coherent sheaves, we also have that FD∗ restricts to

F : FRepr ( ) → Coh(pt), D∗ k Gm so FD∗ preserves coherence. By definition of group homology, see [11] chapter 16, as right derived i i functors of the invariants functor, we now have that R FD∗ is H (Gm, −). We use the following vanishing result for the group cohomology.

Lemma 4.27. For all representations (V, ρ) of Gm of weight n, the group cohomology is given by ( V for n = 0 and i = 0 Hi( ,V ) = Gm 0 otherwise.

Proof. Suppose V is a weight n representation. For i = 0, we have

∗ 0 Gm(k) k H (Gm,V ) = V (k) = V .

If Gm acts trivially, so if n = 0, this results in V and else this is the 0 vector 0 space. For general representations V we then have H (Gm,V ) = V0 the weight 0 subrepresentation of V . For i ≥ 0, we use the classification in Corollary 2.16 to conclude that Gm i is reductive, and we may use Proposition 16.16 in [11] giving H (Gm,V ) = 0 for all i > 0 and all representations V .

Proof of 4.23. Let

O ∗ O ∗ ⊗ni
α = L (⊗a∈ARπ∗φ Va) ⊗b∈B evi [Wb] ⊗ [Ti ]

48 ∗ ⊗n be an admissible complex. Firstly, by Lemma 4.21, L = (det Rπ∗φ Gm1) for n ∈ Z \{0} has weight w(L) = n(D + 1), which tends to ±∞ for i D → ±∞. Hence for all L, R FD∗L will vanish for D → ±∞ by Lemma 4.27. If the other complexes, evaluation bundles and descendant bundles that are tensored with L in an admissible complex are bounded in weight (from above and below), then we still have that the weight of α will go to ±∞ (dependent on growth of the weight of L) for D → ±∞. An example of the distribution of weights for D ∈ Z is illistrated in Figure 4.4. L ⊗nj For a finite-dimensional representation V = j∈J Gm1 of Gm, con- ∗ i ∗ sider the complex Rπ∗φ V . For each R π∗φ V , we have by Remark 4.15 that we get direct sums of weight nj representations when i = 0, 1. So in particular weights for such a complex are bounded. ∗ Let W be a weight w(W ) representation. For evaluation bundles evi [W ], by remark 4.4, we have evi ◦jD = idBGm and so on the degree D component ∗ evi [W ] has weight w(W ). ni For descendant bundles, we can also tensor with [Ti ] for some integer n . Consider σ∗Ω . Because we have the following pullback diagram i i C/Me

q 1 C Pk 0 σi σi F Me k, ∗ we have for the relative differentials that Ω =∼ q Ω 1 and so C/Me Pk/k

0∗ 0∗ ∗ ∗ ∗ σ Ω =∼ σ q Ω 1 =∼ F σ Ω 1 . i C/Me i Pk/k i Pk/k

∗ As σ Ω 1 is a line bundle on the field k, this gives the trivial sheaf, and i Pk/k so σ0∗Ω = O . Hence the classes [T ni ] always have weight 0 on each D. i C/Me Me i Hence, w(α) goes to ∞ or −∞ for D 7→ ±∞, and so by by Lemma 4.27, the pushforward along FD will all vanish but for finitely many D. Therefore, the pushforward RF∗α is coherent.

49 weight

L

∗ #(Lj) Rπ∗φ V

∗ w(W1) evi W1

D → −∞ D = 0 D → ∞

∗ w(W2) evi W2

Figure 4: Example distribution of weights of components of an admissible complex.

4.5 Gromov-Witten invariants for Me 0,3(BGm) Theorem 4.23 shows that the K-theoretic pushforward exists, but we may even calculate the numerical Gromov-Witten invariant for an admissible complex, given the following data. ∗ ⊗−n Let n ∈ Z \{0} be given and consider L = (det Rπ∗φ Gm1) . Be- cause any representation is a direct sum of one-dimensional representations, and the following constructions respect direct sums, let Va,Wb be one- dimensional weight va resp. wb representations of Gm. Finally, let nj ∈ Z be given, and consider the complex

O ∗ O  ∗ ⊗nj  α = L (⊗a∈ARπ∗φ Va) ⊗b∈B evj [Wb] ⊗ [Tj ] .

By definition of the K-theoretic pushforward, we compute

X i i F∗[α] = (−1) [R F∗α]. i

i For D ∈ Z, we know R FD∗ is only non-zero if i = 0 and we have a weight

50 0 representation, by lemma 4.27. Thus we want to compute 0 M 0 R F∗(α) = R FD∗(α). D Recall that on the degree D component, the components of α are the fol- lowing representations: • The admissible line bundle L is a one-dimensional weight n(D + 1) representation of Gm. 0 ∗ 0 1 • If va · D > −1, then R π∗φ Va is a dim H (Pk, O(vaD))-dimensional 1 ∗ weight va representation. If va · D < −1, then R π∗φ Va is a weight 1 1 va representation of dimension dim H (Pk, O(vaD)). In other cases, i ∗ R π∗φ Va is a zero-dimensional representation. For convenience, write 0 1 0 1 h0(vaD) = dim H (Pk, O(vaD)) and h1(vaD) = dim H (Pk, O(vaD)); we computed these dimensions explicitely in (4) in Example 4.14. Then X i i ∗ 0 ∗ 1 ∗ (−1) [R π∗φ Va] = [R π∗φ Va] − [R π∗φ Va] i is a h0(vaD)-dimensional representation minus a h1(vaD)-dimensional representation.

∗ • evi [Wb] is a one-dimensional weight wb representation, and

⊗nj • [Ti ] is a one-dimensional weight 0 representation. Only for D such that the sum of these weights X X n(D + 1) + va + wb

a:vaD6=−1 b equals 0, FD∗ can possibly not give 0. Such D exists if we have P P −n − va − wb D := a:vaD6=−1 b ∈ . 0 n Z 0 0 Because R FD0 ∗ sends a weight 0 representation V to H (Gm,V ) = V , and the isomorphism from K0(pt) to Z sends a vector space to its dimension, we consider the dimension of the above representations. For k-vector spaces V,W , the tensor product satisfies dim(V ⊗ W ) = dim(V ) dim(W ), and so when tensoring we multiply dimensions. Applying this to the above list of representations, we obtain the following result. Theorem 4.28. The numerical Gromov-Witten invariant of α is given by Y Y h0(vaD0) · −h1(vaD0).

a:vaD>−1 a:vaD<−1 if D0 ∈ Z, and the numerical Gromov-Witten invariant equals 0 otherwise.

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