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MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY
VICTORIA HOSKINS Abstract In this course, we study moduli problems in algebraic geometry and the construction of moduli spaces using geometric invariant theory. We start by giving the definitions of coarse and fine moduli spaces, with an emphasis on examples. We then explain how to construct group quotients in algebraic geometry via geometric invariant theory. Finally, we apply these techniques to construct moduli spaces of projective hypersurfaces and moduli spaces of semistable vector bundles on a smooth projective curve.
- Introduction Contents
- Notation and conventions
- Acknowledgements
- Moduli problems.
- 2.1. Functors of points
- 2.2. Moduli problem
- 2.3. Fine moduli spaces
- 2.4. Pathological behaviour.
- 2.5. Coarse moduli spaces
- 2.6. The construction of moduli spaces
- Algebraic group actions and quotients
- 3.1. Affine Algebraic groups
- 3.2. Group actions
- 3.3. Orbits and stabilisers
- 3.4. First notions of quotients
- 3.5. Second notions of quotient
- 3.6. Moduli spaces and quotients
- Affine Geometric Invariant Theory
- 4.1. Hilbert’s 14th problem
- 4.2. Reductive groups
- 4.3. Nagata’s theorem
- 4.4. Reynolds operators
- 4.5. Construction of the affine GIT quotient
- 4.6. Geometric quotients on open subsets
- Projective GIT quotients
- 5.1. Construction of the projective GIT quotient
- 5.2. A description of the k-points of the GIT quotient
- 5.3. Linearisations
- 5.4. Projective GIT with respect to an ample linearisation
- 5.5. GIT for general varieties with linearisations
- Criteria for (semi)stability
- 6.1. A topological criterion
- 6.2. The Hilbert–Mumford Criterion
- 6.3. The Hilbert–Mumford Criterion for ample linearisations
- 6.4. Proof of the Fundamental Theorem in GIT
- Moduli of projective hypersurfaces
- 7.1. The moduli problem
- 7.2. Singularities of hypersurfaces
2 VICTORIA HOSKINS
7.3. The Hilbert–Mumford criterion for hypersurfaces.............................. 53 7.4. Binary forms of degree d....................................................... 53 7.5. Plane cubics................................................................... 55
- Moduli of vector bundles on a curve............................................... 60 Convention:......................................................................... 60 8.1. An overview of sheaf cohomology.............................................. 61 8.2. Line bundles and divisors on curves............................................ 62 8.3. Serre duality and the Riemann-Roch Theorem................................. 63 8.4. Vector bundles and locally free sheaves........................................ 64 8.5. Semistability.................................................................. 66 8.6. Boundedness of semistable vector bundles...................................... 68 8.7. The Quot scheme.............................................................. 69 8.8. GIT set up for construction of the moduli space................................ 74 8.9. Analysis of semistability....................................................... 76 8.10. Construction of the moduli space............................................. 82 References............................................................................. 86
- Introduction
In this course, we study moduli problems in algebraic geometry and constructions of moduli spaces using geometric invariant theory. A moduli problem is essentially a classification problem: we want to classify certain geometric objects up to some notion of equivalence (key examples are vector bundles on a fixed variety up to isomorphism or hypersurfaces in Pn^ up to projective transformations). We are also interested in understanding how these objects deform in families and this information is encoded in a moduli functor. An ideal solution to a moduli problem is a (fine) moduli space, which is a scheme that represents this functor. However, there are many simple moduli problems which do not admit such a solution. Often we must restrict our attention to well-behaved objects to construct a moduli space. Typically the construction of moduli spaces is given by taking a group quotient of a parameter space, where the orbits correspond to the equivalence classes of objects. Geometric invariant theory (GIT) is a method for constructing group quotients in algebraic geometry and it is frequently used to construct moduli spaces. The core of this course is the construction of GIT quotients. Eventually we return to our original motivation of moduli problems and construct moduli spaces using GIT. We complete the course by constructing moduli spaces of projective hypersurfaces and moduli spaces of (semistable) vector bundles over a smooth complex projective curve. Let us recall the quotient construction in topology: given a group G acting on a topological space X, we can give the orbit space X/G := {G · x : x ∈ X} the quotient topology, so that the quotient map π : X → X/G is continuous. In particular, π gives a quotient in the category of topological spaces. More generally, we can suppose G is a Lie group and X has the structure of a smooth manifold. In this case, the quotient X/G will not always have the structure of a smooth manifold (for example, the presence of non-closed orbits, usually gives a non-Hausdorff quotient). However, if G acts properly and freely, then X/G has a smooth manifold structure, such that π is a smooth submersion. In this course, we are interested in actions of an affine algebraic group G (that is, an affine scheme with a group structure such that multiplication and inversion are algebraic morphisms). More precisely, we’re interested in algebraic G-actions on an algebraic variety (or scheme of finite type) X over an algebraically closed field k. As most affine groups are non-compact, their actions typically have some non-closed orbits. Consequently, the topological quotient X/G will not be Hausdorff. However one could also ask whether we should relax the idea of having an orbit space, in order to get a quotient with better geometrical properties. More precisely, we ask for a categorical quotient in the category of finite type k-schemes; that is, a G-invariant morphism π : X → Y which is universal (i.e., every other G-invariant morphism X → Z factors
4 VICTORIA HOSKINS
Acknowledgements. First and foremost, I would like to thank Frances Kirwan for introducing me to geometric invariant theory. These lectures notes were written for a masters course at the Freie Universit¨at Berlin in the winter semester of 2015 and I am grateful to have had some excellent students who actively participated in the course and engaged with the topic: thank you to Anna-Lena, Christoph, Claudius, Dominic, Emmi, Fei, Felix, Jennifer, Koki, Maciek, Maik, Maksymilian, Markus, and Vincent. Thanks also to Eva Mart´ınez for all her hard work running the exercise sessions and to the students for persevering with the more challenging exercises. I am very grateful to Giangiacomo Sanna for many useful comments and suggestions on the script. Finally, I would especially like to thank Simon Pepin Lehalleur for numerous helpful discussions and contributions, and for his encyclopaedic knowledge of algebraic groups.
- Moduli problems
2.1. Functors of points. In this section, we will make use of some of the language of category theory. We recall that a morphism of categories C and D is given by a (covariant) functor F : C → D, which associates to every object C ∈ C an object F (C) ∈ D and to each morphism f : C → C′^ in C a morphism F (f ) : F (C) → F (C′) in D such that F preserves identity morphisms and composition. A contravariant functor F : C → D reverses arrows: so F sends f : C → C′^ to F (f ) : F (C′) → F (C). The notion of a morphism of (covariant) functors F, G : C → D is given by a natural trans- formation η : F → G which associates to every object C ∈ C a morphism ηC : F (C) → G(C) in D which is compatible with morphisms f : C → C′^ in C, i.e. we have a commutative square
F (C) F (f )
ηC (^) // G(C) G(f ) F (C′) (^) η C′
/ / G(C
We note that if F and G were contravariant functors, the vertical arrows in this square would be reversed. If ηC is an isomorphism in D for all C ∈ C, then we call η a natural isomorphism or simply an isomorphism of functors.
Remark 2.1. The focus of this course is moduli problems, rather than category theory and so we are doing naive category theory (in the sense that we allow the objects of a category to be a class). This is analogous to doing naive set theory without a consistent axiomatic approach. However, for those interested in category theory, this can all be handled in a consistent manner, where one pays more careful attention to the size of the set of objects. One approach to this more formal category theory can be found in the book of Kashiwara and Schapira [18]. Strictly speaking, in this case, one should work with the category of ‘small’ sets.
Let Set denote the category of sets and let Sch denote the category of schemes (of finite type over k).
Definition 2.2. The functor of points of a scheme X is a contravariant functor hX := Hom(−, X) : Sch → Set from the category of schemes to the category of sets defined by
hX (Y ) := Hom(Y, X) hX (f : Y → Z) := hX (f ) : hX (Z) → hX (Y ) g 7 → g ◦ f.
Furthermore, a morphism of schemes f : X → Y induces a natural transformation of functors hf : hX → hY given by
hf,Z : hX (Z) → hY (Z) g 7 → f ◦ g.
MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 5
Contravariant functors from schemes to sets are called presheaves on Sch and form a cate- gory, with morphisms given by natural transformations; this category is denoted Psh(Sch) := Fun(Schop, Set), the category of presheaves on Sch. The above constructions can be phrased as follows: there is a functor h : Sch → Psh(Sch) given by
X 7 → hX (f : X → Y ) 7 → hf : hX → hY.
In fact, there is nothing special about the category of schemes here. So for any category C, there is a functor h : C → Psh(C).
Example 2.3. For a scheme X, we have that hX (Spec k) := Hom(Spec k, X) is the set of k-points of X and, for another scheme Y , we have that hX (Y ) is the set of Y -valued points of X. Let X = A^1 be the affine line; then the functor of points hA 1 associates to a scheme Y the set of functions on Y (i.e. morphisms Y → A^1 ). Similarly, for the scheme Gm = A^1 − { 0 }, the functor hA 1 associates to a scheme Y the set of invertible functions on Y.
Lemma 2.4 (The Yoneda Lemma). Let C be any category. Then for any C ∈ C and any presheaf F ∈ Psh(C), there is a bijection
{natural transformsations η : hC → F } ←→ F (C).
given by η 7 → ηC (IdC ).
Proof. Let us first check that this is surjective: for an object s ∈ F (C), we define a natural transformation η = η(s) : hC → F as follows. For C′^ ∈ C, let ηC′^ : hC (C′) → F (C′) be the morphism of sets which sends f : C′^ → C to F (f )(s) (recall that F (f ) : F (C) → F (C′)). This is compatible with morphisms and, by construction, ηC (idC ) = F (idC )(s) = s. For injectivity, suppose we have natural transformations η, η′^ : hC → F such that ηC (IdC ) = η C′ (IdC ). Then we claim η = η′; that is, for any C′^ in C, we have ηC′^ = η′ C′ : hC (C′) → F (C′). Let g : C′^ → C, then as η is a natural transformation, we have a commutative square
hC (C) hC (g)
ηC (^) // F (C) F (g) hC (C′) (^) ηC′ //F (C′).
It follows that
(F (g) ◦ ηC )(idC ) = (ηC′ ◦ hC (g))(IdC ) = ηC′ (g)
and similarly, as η′^ is a natural transformation, that (F (g) ◦ η C′ )(idC ) = η C′′ (g). Hence
ηC′ (g) = F (g)(ηC (idC )) = F (g)(η C′ (idC )) = η C′′ (g)
as required.
The functor h : C → Psh(C) is called the Yoneda embedding, due to the following corollary.
Corollary 2.5. The functor h : C → Psh(C) is fully faithful.
Proof. We recall that a functor is fully faithful if for every C, C′^ in C, the morphism
HomC (C, C′) → HomPsh(C)(hC , hC′^ )
is bijective. This follows immediately from the Yoneda Lemma if we take F = hC′^.
Exercise 2.6. Show that if there is a natural isomorphism hC → h′ C , then there is a canonical isomorphism C → C′.
The presheaves in the image of the Yoneda embedding are known as representable functors.
Definition 2.7. A presheaf F ∈ Psh(C) is called representable if there exists an object C ∈ C and a natural isomorphism F ∼= hC.
MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 7
where πS : X × S → S. For the second equivalence relation, since L → S is locally trivial, there is a cover Si of S such that F|X×Si ∼= G|X×Si. It turns out that the second notion of equivalence offers the extra flexibility we will need in order to construct moduli spaces.
Example 2.13. Let A consist of 4 ordered distinct points (p 1 , p 2 , p 3 , p 4 ) on P^1. We want to classify these quartuples up to the automorphisms of P^1. We recall that the automorphism group of P^1 is the projective linear group PGL 2 , which acts as M¨obius transformations. We define our equivalence relation by (p 1 , p 2 , p 3 , p 4 ) ∼ (q 1 , q 2 , q 3 , q 4 ) if there exists an automorphisms f : P^1 → P^1 such that f (pi) = qi for i = 1,... , 4. We recall that for any 3 distinct points (p 1 , p 2 , p 3 ) on P^1 , there exists a unique M¨obius transformation f ∈ PGL 2 which sends (p 1 , p 2 , p 3 ) to (0, 1 , ∞) and the cross-ratio of 4 distinct points (p 1 , p 2 , p 3 , p 4 ) on P^1 is given by f (p 4 ) ∈ P^1 − { 0 , 1 , ∞}, where f is the unique M¨obius transformation that sends (p 1 , p 2 , p 3 ) to (0, 1 , ∞). Therefore, we see that the set A/ ∼ is in bijection with the set of k-points in the quasi-projective variety P^1 − { 0 , 1 , ∞}. In fact, we can naturally speak about families of 4 distinct points on P^1 over a scheme S: this is given by a proper flat morphism π : X → S such that the fibres π−^1 (s) ∼= P^1 are smooth rational curves and 4 disjoint sections (σ 1 ,... , σ 4 ) of π. We say two families (π : X → S, σ 1 ,... , σ 4 ) and (π′^ : X ′^ → S, σ 1 ′,... , σ 4 ′) are equivalent over S if there is an isomorphism f : X → X ′^ over S (i.e. π = π′^ ◦ f ) such that f ◦ σi = σ′ i. There is a tautological family over the scheme S = P^1 −{ 0 , 1 , ∞}: let π : P^1 −{ 0 , 1 , ∞}×P^1 → S = P^1 − { 0 , 1 , ∞} be the projection map and choose sections (σ 1 (s) = 0, σ 2 (s) = 1, σ 3 (s) = ∞, σ 4 (s) = s). It turns out that this family over P^1 −{ 0 , 1 , ∞} encodes all families parametrised by schemes S (in the language to come, U is a universal family and P^1 −{ 0 , 1 , ∞} is a fine moduli space).
Exercise 2.14. Define an analogous notion for families of n ordered distinct points on P^1 and let the corresponding moduli functor be denoted M 0 ,n (this is the moduli functor of n ordered distinct points on the curve P^1 of genus 0). For n = 3, show that M 0 , 3 (Spec k) is a single element set and so is in bijection with the set of k-points of Spec k. Furthermore, show there is a tautological family over Spec k.
2.3. Fine moduli spaces. The ideal situation is when there is a scheme that represents our given moduli functor.
Definition 2.15. Let M : Sch → Set be a moduli functor; then a scheme M is a fine moduli space for M if it represents M.
Let’s carefully unravel this definition: M is a fine moduli space for M if there is a natural isomorphism η : M → hM. Hence, for every scheme S, we have a bijection
ηS : M(S) := {families over S}/ ∼S ←→ hM (S) := {morphisms S → M }.
In particular, if S = Spec k, then the k-points of M are in bijection with the set A/ ∼. Furthermore, these bijections are compatible with morphisms T → S, in the sense that we have a commutative diagram
M(S) M(f )
ηS (^) // hM (S) hM (f ) M(T ) (^) ηT //hM (T ).
The natural isomorphism η : M → hM determines an element U = η M−^1 (idM ) ∈ M(M ); that is, U is a family over M (up to equivalence).
Definition 2.16. Let M be a fine moduli space for M; then the family U ∈ M(M ) corre- sponding to the identity morphism on M is called the universal family.
This family is called the universal family, as any family F over a scheme S (up to equivalence) corresponds to a morphism f : S → M and, moreover, as the families f ∗U and F correspond
8 VICTORIA HOSKINS
to the same morphism idM ◦ f = f , we have
f ∗U ∼S F;
that is, any family is equivalent to a family obtained by pulling back the universal family.
Remark 2.17. If a fine moduli space for M exists, it is unique up to unique isomorphism: that is, if (M, η) and (M ′, η′) are two fine moduli spaces, then they are related by unique isomorphisms η′ M ((ηM )−^1 (IdM )) : M → M ′^ and ηM ′ ((η′ M ′ )−^1 (IdM ′ )) : M ′^ → M.
We recall that a presheaf F : Sch → Set is said to be a sheaf in the Zariski topology if for every scheme S and Zariski cover {Si} of S, the natural map
{f ∈ F (S)} −→ {(fi ∈ F (Si))i : fi|Si∩Sj = fj |Sj ∩Si for all i, j}
is a bijection. A presheaf is called a separated presheaf if these natural maps are injective.
Exercise 2.18.
(1) Show that the functor of points of a scheme is a sheaf in the Zariski topology. In particular, deduce that for a presheaf to be representable it must be a sheaf in the Zariski topology. (2) Consider the moduli functor of vector bundles over a fixed scheme X, where we say two families E and F are equivalent if and only if they are isomorphic. Show that the corresponding moduli functor fails to be a separable presheaf (it may be useful to consider the second equivalence relation we introduced for families of vector bundles in Exercise 2.12).
Example 2.19. Let us consider the projective space Pn^ = Proj k[x 0 ,... , xn]. This variety can be interpreted as a fine moduli space for the moduli problem of lines through the origin in V := An+1. To define this moduli problem carefully, we need to define a notion of families and equivalences of families. A family of lines through the origin in V over a scheme S is a line bundle L over S which is a subbundle of the trivial vector bundle V × S over S (by subbundle we mean that the quotient is also a vector bundle). Then two families are equivalent if and only if they are equal. Over Pn, we have a tautological line bundle OPn (−1) ⊂ V × Pn, whose fibre over p ∈ Pn^ is the corresponding line in V. This provides a tautological family of lines over Pn. The dual of the tautological line bundle is the line bundle OPn^ (1), known as the Serre twisting sheaf. The important fact we need about OPn^ (1) is that it is generated by the global sections x 0 ,... , xn. Given any morphism of schemes f : S → Pn, the line bundle f ∗OPn^ (1) is generated by the global sections f ∗(x 0 ),... , f ∗(xn). Hence, we have a surjection On S+1 → f ∗OPn (1). For locally free sheaves, pull back commutes with dualising and so
f ∗OPn^ (−1) ∼= (f ∗OPn^ (1))∨.
Dually the above surjection gives an inclusion L := f ∗OPn^ (−1) → On S+1 = V × S which determines a family of lines in V over S. Conversely, let L ⊂ V × S be a family of lines through the origin in V over S. Then, dual to this inclusion, we have a surjection q : V ∨^ × S → L∨. The vector bundle V ∨^ × S is generated by the global sections σ 0 ,... , σn corresponding to the dual basis for the standard basis on V. Since q is surjective, the dual line bundle L∨^ is generated by the global sections q ◦ σ 0 ,... , q ◦ σn. In particular, there is a unique morphism f : S → Pn^ given by
s 7 → [q ◦ σ 0 (s) : · · · : q ◦ σn(s)]
such that f ∗OPn^ (−1) = L ⊂ V × S (for details, see [14] II Theorem 7.1). Hence, there is a bijective correspondence between morphisms S → Pn^ and families of lines through the origin in V over S. In particular, Pn^ is a fine moduli space and the tautological family is a universal family. The keen reader may note that the above calculations suggests we should rather think of Pn^ as the space of 1-dimensional quotient spaces of a n + 1-dimensional vector space (a convention that many algebraic geometers use).
10 VICTORIA HOSKINS
We claim there is no family F over a scheme S with the property that for any rank 2 degree 0 vector bundle E on P^1 , there is a k-point s ∈ S such that F|s ∼= E. Suppose such a family F over a scheme S exists. For each n ∈ N, we have a rank 2 degree 0 vector bundle OP 1 (n) ⊕ OP 1 (−n) (in fact, by Grothendieck’s Theorem classifying vector bundles on P^1 , every rank 2 degree 0 vector bundle on P^1 has this form). Furthermore, we have
dim H^0 (P^1 , OP 1 (n) ⊕ OP 1 (−n)) = dimk(k[x 0 , x 1 ]n ⊕ k[x 0 , x 1 ]−n) =
2 if n = 0, n + 1 if n ≥ 1.
Consider the subschemes Sn := {s ∈ S : dim H^0 (P^1 , Fs) ≥ n} of S, which are closed by the semi-continuity theorem (see [14] III Theorem 12.8). Then we obtain a decreasing chain of closed subschemes S = S 2 % S 3 % S 4 % ....
each of which is distinct as OP 1 (n) ⊕ OP 1 (−n) ∈ Sn+1 − Sn+2. The existence of this chain contradicts the fact that S is Noetherian (recall that for us scheme means scheme of finite type over k). In particular, the moduli problem of vector bundles of rank 2 and degree 0 is unbounded. In fact, we also see the jump phenomena: there is a family F of rank 2 degree 0 vector bundles over A^1 = Spec k[s] such that
Fs =
O⊕ P 12 s 6 = 0 OP 1 (1) ⊕ OP 1 (−1) s = 0.
To construct this family, we note that
Ext^1 (OP 1 (1), OP 1 (−1)) ∼= H^1 (P^1 , OP 1 (−2)) ∼= H^0 (P^1 , OP 1 )∗^ ∼= k
by Serre duality. Therefore, there is a family of extensions F over A^1 of OP 1 (1) by OP 1 (−1) with the desired property.
In both cases there is no fine moduli space for this problem. To solve these types of phenom- ena, one usually restricts to a nicer class of objects (we will return to this idea later on).
Example 2.23. We can see more directly that there is no fine moduli space for Endn. Suppose M is a fine moduli space. Then we have a bijection between morphisms S → M and families over S up to equivalence. Choose any n × n matrix T , which determines a point m ∈ M. Then for S = P^1 we have that the trivial families (O Pn 1 , T ) and (O Pn 1 ⊗ OP 1 (1), T ⊗ IdOP 1 (1))
are non-equivalent families which determine the same morphism P^1 → M , namely the constant morphism to the point m.
2.5. Coarse moduli spaces. As demonstrated by the above examples, not every moduli func- tor has a fine moduli space. By only asking for a natural transformation M → hM which is universal and a bijection over Spec k (so that the k-points of M are in bijection with the equivalence classes A/ ∼), we obtain a weaker notion of a coarse moduli space.
Definition 2.24. A coarse moduli space for a moduli functor M is a scheme M and a natural transformation of functors η : M → hM such that
(a) ηSpec k : M(Spec k) → hM (Spec k) is bijective. (b) For any scheme N and natural transformation ν : M → hN , there exists a unique morphism of schemes f : M → N such that ν = hf ◦ η, where hf : hM → hN is the corresponding natural transformation of presheaves.
Remark 2.25. A coarse moduli space for M is unique up to unique isomorphism: if (M, η) and (M ′, η′) are coarse moduli spaces for M, then by Property (b) there exists unique morphisms f : M → M ′^ and f ′^ : M ′^ → M such that hf and hf ′^ fit into two commutative triangles:
hM hf
M
oo^ η
} }^ η′
η′^ //
η (^) ""
hM ′ hf ′ hM ′ hM.
MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 11
Since η = hf ′ ◦ hf ◦ η and η = hidM ◦ η, by uniqueness in (b) and the Yoneda Lemma, we have f ′^ ◦ f = idM and similarly f ◦ f ′^ = idM ′.
Proposition 2.26. Let (M, η) be a coarse moduli space for a moduli problem M. Then (M, η) is a fine moduli space if and only if
(1) there exists a family U over M such that ηM (U) = idM , (2) for families F and G over a scheme S, we have F ∼S G ⇐⇒ ηS (F) = ηS (G).
Proof. Exercise.
Lemma 2.27. Let M be a moduli problem and suppose there exists a family F over A^1 such that Fs ∼ F 1 for all s 6 = 0 and F 0 F 1. Then for any scheme M and natural transformation η : M → hM , we have that ηA 1 (F) : A^1 → M is constant. In particular, there is no coarse moduli space for this moduli problem.
Proof. Suppose we have a natural transformation η : M → hM ; then η sends the family F over A^1 to a morphism f : A^1 → M. For any s : Spec k → A^1 , we have that f ◦ s = ηSpec k(Fs) and, for s 6 = 0, Fs = F 1 ∈ M(Spec k), so that f |A (^1) −{ 0 } is a constant map. Let m : Spec k → M be the point corresponding to the equivalence class for F 1 under η. Since the k-valued points of M are closed (recall M is a scheme of finite type over an algebraically closed field), their preimages under morphisms must also be closed. Then, as A^1 − { 0 } ⊂ f −^1 (m), the closure A^1 of A^1 − { 0 } must also be contained in f −^1 (m); that is, f is the constant map to the k-valued point m of M. In particular, the map ηSpec k : M(Spec k) → hM (Spec k) is not a bijection, as F 0 6 = F 1 in M(Spec k), but these non-equivalent objects correspond to the same k-point m in M.
In particular, the moduli problems of Examples 2.22 and 2.21 do not even admit coarse moduli spaces.
2.6. The construction of moduli spaces. The construction of many moduli spaces follows the same general pattern.
(1) Fix any discrete invariants for our objects - here the invariants should be invariant under the given equivalence relation (for example, for isomorphism classes of vector bundles on a curve, one may fix the rank and degree). (2) Restrict to a reasonable class of objects which are bounded (otherwise, we can’t find a coarse moduli space). Usually one restricts to a class of stable objects which are better behaved and bounded. (3) Find a family F over a scheme P with the local universal property (i.e. locally any other family is equivalent to a pullback of this family - see below). We call P a parameter space, as the k-points of P surject onto A/ ∼; however, this is typically not a bijection. (4) Find a group G acting on P such that p and q lie in the same G-orbit in P if and only if Fp ∼ Fq. Then we have a bijection P (k)/G ∼= A/ ∼. (5) Typically this group action is algebraic (see Section 3) and by taking a quotient, we should obtain our moduli space. The quotient should be taken in the category of schemes (in terminology to come, it should be a categorical quotient) and this is done using Mumford’s Geometric Invariant Theory.
Definition 2.28. For a moduli problem M, a family F over a scheme S has the local universal property if for any other family G over a scheme T and for any k-point t ∈ T , there exists a neighbourhood U of t in T and a morphism f : U → S such that G|U ∼U f ∗F.
In particular, we do not require the morphism f to be unique. We note that, for such a family to exist, we need our moduli problem to be bounded.
- Algebraic group actions and quotients In this section we consider group actions on algebraic varieties and also describe what type of quotients we would like to have for such group actions.
MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 13
algebraic group is separated. Hence, in characteristic zero, the notion of affine algebraic group and affine group variety coincide. (4) In the definition of homomorphisms, we only require a compatibility with the group law m; it turns out that the compatibility for the identity and group inversion is then automatic. This is well known in the case of homomorphisms of abstract groups, and the algebraic case can then be deduced by applying the Yoneda lemma. (5) For the definition of a quotient group, the condition that the homomorphism is flat is only needed in positive characteristic, as in characteristic zero this morphism is already smooth (this follows from the Theorem of Cartier mentioned above and the fact that the kernel of a homomorphism of smooth group schemes is smooth; see [22] Proposition 1.48)
Example 3.3. Many of the groups that we are already familiar with are affine algebraic groups.
(1) The additive group Ga = Spec k[t] over k is the algebraic group whose underlying variety is the affine line A^1 over k and whose group structure is given by addition: m∗(t) = t ⊗ 1 + 1 ⊗ t and i∗(t) = −t. Let us indicate how to show these operations satisfy the group axioms. We only prove the associativity, the other axioms being similar and easier. We have to show that (m∗^ ⊗ id) ◦ m∗^ = (id ⊗m∗) ◦ m∗^ : k[t] → k[t] ⊗ k[t] ⊗ k[t]. This is a map of k-algebras, so it is enough to check it for t. We have ((m∗^ ⊗ id) ◦ m∗)(t) = (m∗^ ⊗ id)(t ⊗ 1 + 1 ⊗ t) = t ⊗ 1 ⊗ 1 + 1 ⊗ t ⊗ 1 + 1 ⊗ 1 ⊗ t and similarly ((id ⊗m∗) ◦ m∗)(t) = t ⊗ 1 ⊗ 1 + 1 ⊗ t ⊗ 1 + 1 ⊗ 1 ⊗ t which completes the proof. For a k-algebra R, we have Ga(R) = (R, +); this justifies the name of the ‘additive group’. (2) The multiplicative group Gm = Spec k[t, t−^1 ] over k is the algebraic group whose under- lying variety is the A^1 − { 0 } and whose group action is given by multiplication: m∗(t) = t ⊗ t and i∗(t) = t−^1. For a k-algebra R, we have Gm(R) = (R×, ·); hence, the name of the ‘multiplicative group’. (3) The general linear group GLn over k is an open subvariety of An
2 cut out by the condition that the determinant is non-zero. It is an affine variety with coordinate ring k[xij : 1 ≤ i, j ≤ n]det(xij ). The co-group operations are defined by:
m∗(xij ) =
∑^ n
k=
xik ⊗ xkj and i∗(xij ) = (xij )− ij^1
where (xij )− ij^1 is the regular function on GLn given by taking the (i, j)-th entry of the inverse of a matrix. For a k-algebra R, the group GLn(R) is the group of invertible n × n matrices with coefficients in R, with the usual matrix multiplication. (4) More generally, if V is a finite-dimensional vector space over k, there is an affine algebraic group GL(V ) which is (non-canonically) isomorphic to GLdim(V ). For a k-algebra R, we have GL(V )(R) = AutR(V ⊗k R). (5) Let G be a finite (abstract) group. Then G can be naturally seen as an algebraic group Gk over k as follows. The group operations on G make the group algebra k[G] into a Hopf algebra over k, and Gk := Spec(k[G]) is a 0-dimensional variety whose points are naturally identified with elements of G. (6) Let n ≥ 1. Put μn := Spec k[t, t−^1 ]/(tn^ − 1) ⊂ Gm, the subscheme of n-roots of unity. Write I for the ideal (tn^ − 1) of R := k[t, t−^1 ]. Then m∗(tn^ − 1) = tn^ ⊗ tn^ − 1 ⊗ 1 = (tn^ − 1) ⊗ tn^ + 1 ⊗ (tn^ − 1) ∈ I ⊗ R + R ⊗ I
14 VICTORIA HOSKINS
which implies that μn is an algebraic subgroup of Gm. If n is different from char(k), the polynomial Xn^ − 1 is separable and there are n distinct roots in k. Then the choice of a primitive n-th root of unity in k determines an isomorphism μn ' Z/nZk. If n = char(k), however, we have Xn^ − 1 = (X − 1)n^ in k[X], which implies that the scheme μn is non-reduced (with 1 as only closed point). This is the simplest example of a non-reduced algebraic group.
A linear algebraic group is by definition a subgroup of GLn which is defined by polynomial equations; for a detailed introduction to linear algebraic groups, see [1, 15, 40]. For instance, the special linear group is a linear algebraic group. In particular, any linear algebraic group is an affine algebraic group. In fact, the converse statement is also true: any affine algebraic group is a linear algebraic group (see Theorem 3.9 below). An affine algebraic group G over k determines a group-valued functor on the category of finitely generated k-algebras given by R 7 → G(R). Similarly, for a vector space V over k, we have a group valued functor GL(V ) given by R 7 → AutR (V ⊗k R), the group of R-linear automorphisms. If V is finite dimensional, then GL(V ) is an affine algebraic group.
Definition 3.4. A linear representation of an algebraic group G on a vector space V over k is a homomorphism of group valued functors ρ : G → GL(V ). If V is finite dimensional, this is equivalent to a homomorphism of algebraic groups ρ : G → GL(V ), which we call a finite dimensional linear representation of G.
If G is affine, we can describe a linear representation ρ : G → GL(V ) more concretely in terms of its associated co-module as follows. The natural inclusion GL(V ) → End(V ) and ρ : G → GL(V ) determine a functor G → End(V ), such that the universal element in G(O(G)) given by the identity morphism corresponds to an O(G)-linear endomorphism of V ⊗k O(G), which by the universality of the tensor product is uniquely determined by its restriction to a k-linear homomorphism ρ∗^ : V → V ⊗k O(G); this is the associated co-module. If V is finite dimensional, we can even more concretely describe the associated co-module by considering the group homomorphism G → End(V ) and its corresponding homomorphism of k-algebras O(V ⊗k V ∗) → O(G), which is determined by a k-linear homomorphism V ⊗k V ∗^ → O(G) or equivalently by the co-module ρ∗^ : V → V ⊗k O(G). In particular, a linear representation of an affine algebraic group G on a vector space V is equivalent to a co-module structure on V (for the full definition of a co-module structure, see [23] Chapter 4).
3.2. Group actions.
Definition 3.5. An (algebraic) action of an affine algebraic group G on a scheme X is a morphism of schemes σ : G × X → X such that the following diagrams commute
Spec k × X
e×idX (^) //
∼= ' '
G × X
σ
G × G × X
idG×σ (^) //
mG×idX
G × X
σ X G × X (^) σ //X.
Suppose we have actions σX : G × X → X and σY : G × Y → Y of an affine algebraic group G on schemes X and Y. Then a morphism f : X → Y is G-equivariant if the following diagram commutes
G × X
idG×f (^) //
σX
G × Y
σY X (^) f //Y.
If Y is given the trivial action σY = πY : G×Y → Y , then we refer to a G-equivariant morphism f : X → Y as a G-invariant morphism.
16 VICTORIA HOSKINS
Finally, we claim that ρ : G → Matn×n is a homomorphism of semigroups (recall that a semigroup is a group without inversion, such as matrices under multiplication) i.e. we want to show on the level of k-algebras that we have a commutative square
O(Matn×n)
m∗ Mat (^) //
ρ∗
O(Matn×n) ⊗ O(Matn×n) ρ∗⊗ρ∗ O(G) m∗ G
/ / O(G) ⊗ O(G);
that is, we want to show for the generators xij ∈ O(Matn×n), we have
m∗ G(aij ) = m∗ G(ρ∗(xij )) = (ρ∗^ ⊗ ρ∗)(m∗ Mat(xij )) = (ρ∗^ ⊗ ρ∗)
k
xik ⊗ xkj
k
aik ⊗ akj.
To prove this, we consider the associativity identity mG ◦ (id × mG) = mG ◦ (mG × id) and apply this on the k-algebra level to fi ∈ O(G) to obtain ∑
k,j
aik ⊗ akj ⊗ fj =
j
m∗ G(aij ) ⊗ fj
as desired. Furthermore, as G is a group rather than just a semigroup, we can conclude that the image of ρ is contained in the group GLn of invertible elements in the semigroup Matn×n.
Tori are a basic class of algebraic group which are used extensively to study the structure of more complicated algebraic groups (generalising the use of diagonal matrices to study matrix groups through eigenvalues and the Jordan normal form).
Definition 3.10. Let G be an affine algebraic group scheme over k.
(1) G is an (algebraic) torus if G ∼= Gnm for some n > 0. (2) A torus of G is a subgroup scheme of G which is a torus. (3) A maximal torus of G is a torus T ⊂ G which is not contained in any other torus.
For a torus T , we have commutative groups
X∗(T ) := Hom(T, Gm) X∗(T ) := Hom(Gm, T )
called the character group and cocharacter group respectively, where the morphisms are homo- morphisms of linear algebraic groups. Let us compute X∗(Gm).
Lemma 3.11. The map
θ : Z → X∗(Gm) n 7 → (t 7 → tn)
is an isomorphism of groups.
Proof. Let us first show that this is well defined. Write m∗^ for the comultiplication on O(Gm). Then m∗(tn) = (t⊗t)n^ = tn^ ⊗tn^ shows that θ(n) : Gm → Gm is a morphism of algebraic groups. Since tatb^ = ta+b, θ itself is a morphism of groups. It is clearly injective, so it remains to show surjectivity. Let φ be an endomorphism of Gm. Write φ∗(t) ∈ k[t, t−^1 ] as
|i|<m ait
i. We have m∗(φ∗(t)) =
φ∗(t) ⊗ φ∗(t), which translates into ∑
i
aiti^ ⊗ ti^ =
i,j
aiaj ti^ ⊗ tj^.
From this, we deduce that at most one ai is non-zero, say an. Looking at the compatibility of φ with the unit, we see that necessarily an = 1. This shows that φ = θ(n), completing the proof.
MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 17
For a general torus T , we deduce from the Lemma that the (co)character groups are finite free Z-modules of rank dim T. There is a perfect pairing between these lattices given by composition
< , >: X∗(T ) × X∗(T ) → Z
where < χ, λ >:= χ ◦ λ. An important fact about tori is that their linear representations are completely reducible. We will often use this result to diagonalise a torus action (i.e. choose a basis of eigenvectors for the T -action so that the action is diagonal with respect to this basis).
Proposition 3.12. For a finite dimensional linear representation of a torus ρ : T → GL(V ), there is a weight space decomposition
V ∼=
χ∈X∗(T )
Vχ
where Vχ = {v ∈ V : t · v = χ(t)v ∀t ∈ T } are called the weight spaces and {χ : Vχ 6 = 0} are called the weights of the action.
Proof. To keep the notation simple, we give the proof for T ∼= Gm, where X∗(T ) ∼= Z; the general case can be obtained either by adapting the proof (with further notation) or by induction on the dimension of T. The representation ρ has an associated co-module
ρ∗^ : V → V ⊗k O(Gm) ∼= V ⊗ k[t, t−^1 ].
and the diagram
V (^) ρ //
ρ
V ⊗ k[t, t−^1 ]
id ⊗m∗ V ⊗ k[t, t−^1 ] ρ⊗id
/ / V ⊗ k[t, t−^1 ] ⊗ k[t, t−^1 ]
commutes. From this, it follows easily that, for each integer m, the space
Vm = {v ∈ V : ρ∗(v) = v ⊗ tm}
is a subrepresentation of V. For v ∈ V , we have ρ∗(v) =
m∈Z fm(v)⊗t
m (^) where fm : V → V is a linear map, and because
of the compatibility with the identity element, we find that
v =
m∈Z
fm(v).
If ρ∗(v) =
m∈Z fm(v)^ ⊗^ t
m, then we claim that fm(v) ∈ Vm. From the diagram above ∑
m∈Z
ρ∗(fm(v)) ⊗ tm^ = (ρ∗^ ⊗ Idk[t,t− (^1) ])(ρ∗(v)) = (IdV ⊗ m∗)(ρ∗(v)) =
m∈Z
fm(v) ⊗ tm^ ⊗ tm
and as {tm}m∈Z are linearly independent in k[t, t−^1 ], the claim follows. Let us show that in fact, the fm form a collection of orthogonal projectors onto the subspaces Vm. Using the commutative diagram again, we get ∑
m∈Z
fm(v) ⊗ tm^ ⊗ tm^ =
m,n∈Z
fm(fn(v)) ⊗ tm^ ⊗ tn,
which again by linear independence of the {tm} shows that fm ◦ fn vanishes if m 6 = n and is equal to fn otherwise; this proves that they are orthogonal idempotents. Hence, the Vm are linearly independent and this completes the proof.
This result can be phrased as follows: there is an equivalence between the category of linear representations of T and X∗(T )-graded k-vector spaces. We note that there are only finitely many weights of the T -action, for reasons of dimension.
MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 19
Exercise 3.19. In Examples 3.17 and 3.18, write down the coaction homomorphism explicitly.
Proposition 3.20. Let G be an affine algebraic group acting on a scheme X. For x ∈ X(k), we have dim(G) = dim(Gx) + dim(G · x)
Proof. Since the dimension is a topological invariant of a scheme, we can assume G and X are reduced. The orbit G · x, which we see as a locally closed subscheme of X according to the previous proposition, is reduced by definition. This implies that the morphism σx : G → G · x is flat at every generic point of G · x (every k-scheme is flat over k), hence, by the openess of the flat locus of σx (EGA IV 3 11.1.1), there exists a dense open set U such that σ x− 1 (U ) → U is flat. Using the transitive action of G on G · x (which is well defined because G is reduced), we deduce that σx is flat. Moreover, by definition, the fibre of σx at x is the stabiliser Gx. We can thus apply the dimension formula for fibres of a flat morphism [14, Proposition III.9.5], which yields dim(Gx) = dim(G) − dim(G · x)
as required.
Proposition 3.21. Let G be an affine algebraic group acting on a scheme X by a morphism σ : G×X → X. Then the dimension of the stabiliser subgroup (resp. orbit) viewed as a function X → N is upper semi-continuous (resp. lower-semi-continuous); that is, for every n, the sets
{x ∈ X : dim Gx ≥ n} and {x ∈ X : dim(G · x) ≤ n}
are closed in X.
Proof. Consider the graph of the action
Γ = (prX , σ) : G × X → X × X
and the fibre product P
P ϕ^ // _
X _
∆ G × X Γ^ //X × X, where ∆ : X → X × X is the diagonal morphism; then the k-points of the fibre product P consists of pairs (g, x) such that g ∈ Gx. The function on P which sends p = (g, x) ∈ P to the dimension of Pϕ(p) := ϕ−^1 (ϕ(p)) is upper semi-continuous (cf. [14] III 12.8 or EGA IV 13.1.3); that is, for all n {p ∈ P : dim Pϕ(p) ≥ n}
is closed in P. By restricting to the closed subscheme X ∼= {(e, x) : x ∈ X} ⊂ P , we conclude that the dimension of the stabiliser of x is upper semi-continuous; that is,
{x ∈ X : dim Gx ≥ n}
is closed in X for all n. Using the previous proposition, we deduce the statement for dimensions of orbits.
Lemma 3.22. Let G be an affine algebraic group acting on a scheme X over k.
i) If G is an affine group variety and Y and Z are subschemes of X such that Z is closed, then {g ∈ G : gY ⊂ Z} is closed. ii) If X is a variety, then for any subgroup H ⊂ G the fixed point locus XH^ = {x ∈ X : H · x = x} is closed in X.
Proof. Exercise. (Hint: express these subsets as intersections of preimages of closed subschemes under morphisms associated to the action.)
20 VICTORIA HOSKINS
3.4. First notions of quotients. Let G be an affine algebraic group acting on a scheme X over k. In this section and §3.5, we introduce different types of quotients for the action of G on X; the main references for these sections are [4], [25] and [31]. The orbit space X/G = {G · x : x ∈ X} for the G-action on X, may not always admit the structure of a scheme. Instead we ask for a universal quotient in the category of schemes (of finite type over k).
Definition 3.23. A categorical quotient for the action of G on X is a G-invariant morphism ϕ : X → Y of schemes which is universal; that is, every other G-invariant morphism f : X → Z factors uniquely through ϕ so that there exists a unique morphism h : Y → Z such that f = ϕ ◦ h. Furthermore, if the preimage of each k-point in Y is a single orbit, then we say ϕ is an orbit space.
As ϕ is constant on orbits, it is also constant on orbit closures. Hence, a categorical quotient is an orbit space only if the action of G on X is closed; that is, all the orbits G · x are closed.
Remark 3.24. The categorical quotient has nice functorial properties in the following sense: if ϕ : X → Y is G-invariant and we have an open cover Ui of Y such that ϕ| : ϕ−^1 (Ui) → Ui is a categorical quotient for each i, then ϕ is a categorical quotient.
Exercise 3.25. Let ϕ : X → Y be a categorical quotient of a G-action on X.
i) If X is connected, show that Y is connected. ii) If X is irreducible, show that Y is irreducible. iii) If X is reduced, show that Y is reduced.
Example 3.26. We consider the action of Gm on An^ as in Example 3.18. As the origin is in the closure of every single orbit, any G-invariant morphism An^ → Z must be a constant morphism. Therefore, we claim that the categorical quotient is the structure map ϕ : An^ → Spec k to the point Spec k. This morphism is clearly G-invariant and any other G-invariant morphism f : An^ → Z is a constant morphism to z ∈ Z(k). Therefore, there is a unique morphism z : Spec k → Z such that f = z ◦ ϕ.
We now see the sort of problems that may occur when we have non-closed orbits. In Example 3.18 our geometric intuition tells us that we would ideally like to remove the origin and then take the quotient of Gm acting on An^ −{ 0 }. In fact, we already know what we want this quotient to be: the projective space Pn−^1 = (An^ − { 0 })/Gm which is an orbit space for this action.
3.5. Second notions of quotient. Let G be an affine algebraic group acting on a scheme X over k. The group G acts on the k-algebra O(X) of regular functions on X by
g · f (x) = f (g−^1 · x)
and we denote the subalgebra of invariant functions by
O(X)G^ := {f ∈ O(X) : g · f = f for all g ∈ G}.
Similarly if U ⊂ X is a subset which is invariant under the action of G (that is, g · u ∈ U for all u ∈ U and g ∈ G), then G acts on OX (U ) and we write OX (U )G^ for the subalgebra of invariant functions. The following notion of a good quotient came out of geometric invariant theory; more pre- cisely, we will later see that GIT quotients are good quotients. However, it is clear that many of the properties of a good quotient are desirable. Furthermore, we will soon see that a good quotient is a categorical quotient.
Definition 3.27. A morphism ϕ : X → Y is a good quotient for the action of G on X if
i) ϕ is G-invariant. ii) ϕ is surjective. iii) If U ⊂ Y is an open subset, the morphism OY (U ) → OX (ϕ−^1 (U )) is an isomorphism onto the G-invariant functions OX (ϕ−^1 (U ))G. iv) If W ⊂ X is a G-invariant closed subset of X, its image ϕ(W ) is closed in Y.
