Moduli Spaces in Algebraic Geometry: an Introduction [Lecture notes]

104 Pages·2015·0.798 MB·English

by  Günther Trautmann

Moduli Spaces in Algebraic Geometry
an Introduction
uncompleted preliminary version
Gu¨nther Trautmann
Contents
0. Introduction 3
1. Preliminaries and projective morphisms 3
2. Flatness and Base Change 9
3. Hypersurfaces in P 14
n
4. Representable functors 19
5. Grassmannians 23
6. Flag varieties 31
7. Relative flag varieties 36
8. Hilbert schemes and Quot schemes 42
9. Tangent spaces to Quot schemes 51
10. The Hilbert scheme Hilb(P ,2m+2) 59
2
11. The Quot scheme Quot(2OP ,2m+2) 66
2
12. Semistable sheaves 76
13. Moduli spaces of semistable sheaves, an example 83
14. Good Quotients 93
15. Moduli spaces on P 97
2
16. Moduli spaces of instantons 97
17. Appendix A - Flatness 97
18. Appendix B - Depth and local cohomology 99
1
2
19. Appendix C - Chern classes on P (k) and the Hirzebruch–Riemann–Roch–Theorem100
n
References 103
3
0. Introduction
1. Preliminaries and projective morphisms
varieties 1.1. Schemes and varieties
Throughout these notes k will be a fixed algebraically closed field of characteristic 0. A scheme
over k is supposed to be locally of finite type over Spec(k). This means that any point of the
scheme has an affine open neighbourhood U such that the ring O(U) is a finitely generated
k–algebra, i.e. isomorphic to some k[T ,...,T ]/a, see EGA I, 6.2.1 and 6.5. Any such scheme
1 n
is locally noetherian by definition. We denote by
(Sch/k)
the category of schemes which are locally of finite type or locally algebraic over Spec(k) with
morphisms those of schemes over Spec(k). By EGA I, 6.2.3, the morphisms in (Sch/k) are
locally of finite type. An algebraic scheme over k is a scheme which is of finite type over
Spec(k). This means that it has a finite covering of affine open sets U such that O(U) is a
finitely generated k–algebra. Any such scheme is noetherian. By EGA I, 6.3.4, any morphism
between algebraic schemes over k is also of finite type. We denote by
(AlgSch/k)
the category of these schemes.
Because k is algebraically closed, for any scheme X over k, i.e. X ∈ ob(Sch/k), the set X(k) of
k–valued points is the set of its closed points or its k–rational points, see EGA I, 6.5.3. This
set is moreover very dense. This means that for any subset M ⊂ X we have M = M ∩X(k).
When we restrict the structure sheaf O of X to the set X(k) we get a ringed space
X
(X(k),O ) which is locally isomorphic to the maximal spectrum of a finitely generated
X(k)
algebra k[T ,...,T ]/a. Such a ringed space will be called a variety over k if it is also sep-
1 n
arated. This generalizes in a convenient way the notion of a variety by J.P. Serre, allowing
non–reduced and reducible varieties. Any variety over k in our sense is isomorphic to a variety
(X(k),O ) derived from a separated scheme X over k, see EGA I, Appendix. This can
X(k)
easily be seen by gluing the local affine open sets just in the sense of the spectra instead of the
maximal spectra. In fact X 7→ X(k) gives rise to a functor
(Sch/k) → (Var/k)
sep
which is an equivalence of categories. Here (Var/k) denotes the category of varieties over k just
Ha
defined with morphisms those of locally ringed spaces, see EGA I, Appendix or [14], II, 2.6 in
the case of integral varieties. A variety over k will be called an algebraic variety over k if it can
be covered by finitely many affine open sets, each the maximal spectrum of a finitely generated
k-algebra. Again X 7→ X(k) defines an equivalence of categories
(AlgSch/k) → (AlgVar/k).
sep
As we are not interested in change of fields in the context of moduli spaces, Grassmannians,
flag varieties, Hilbert– and Quot schemes, it is convenient to work in the category (Var/k) of
4
varieties over k, see EGA I, Appendix. In this category the fibre product X × Y becomes the
k
usual product X(k)×Y(k). Most of the theorems, definitions or proofs for schemes over k are
wordly the same for varieties or simpler. Therefore we use the general statements for schemes
and coherent sheaves on them in the category (Var/k) without extra proof.
Examples: (1) Let An be the affine n–space over k which is Spec k[T ,...,T ]. It is a sepa-
k 1 n
rated algebraic scheme over k. Its associated algebraic variety An(k) = An(k) is just kn with
k
the induced structure sheaf O , whose stalks are the localizations of the polynomial ring at
kn
maximal ideals corresponding to points of kn.
(2) Similarly P (k) denotes at the same time the usual projective n–space over k and the
n
algebraic variety of closed points of the scheme P = Proj k[T ,...T ]. It is the set of 1–
nk 0 n
dimensional linear subspaces of kn+1. When V is any (n+1)–dimensional vector space, we get
the variety PV ∼= P (k) of 1–dimensional linear subspaces of V.
n
(3) A projective variety X is an algebraic variety over k which can be embedded as a closed
subvariety into some projective space P (k). It is then endowed with the invertible sheaf O (1)
n X
Ha
which is the pull–back of OPn(1) to X. This is then very ample by definition, see [14], II, 7.
Given any line bundle L on a variety X which is generated by its global sections, we are given
the surjection Γ(X,L)⊗O → L and the dual embedding L∗ ֒→ Γ(X,L)∗ ⊗O . It is shown
X X
in section 5 that this induces a unique morphism X → PΓ(X,L)∗ such that L∗ as a subbundle
is isomorphic to the pullback of the tautological subbundle if Γ(X,L) is finite dimensional.
p
dirim 1.2. Direct images: Given a continuous map X −→ Y of ringed spaces, the direct image
sheaf p F of any O –module is defined as the sheaf U 7→ H0(p−1U,F). The derived functors
∗ X
Rjp then define the higher direct image sheaves Rjp F. They can also directly be defined as
∗ ∗
the sheaves associated to the presheaf U 7→ Hj(p−1U,F) on Y. There is the Leray spectral
sequence
Eij = Hi(Y,Rjp F) ⇒ Hi+j(X,F),
2 ∗
see script ”Spektralsequenzen”. If Rjp F = 0 for j > 0, then this sequence degenerates and
∗
yields isomorphisms
Hj(Y,p F) ∼= Hj(X,F)
∗
for any j. If on the other hand Hi(Y,Rjp F) = 0 for any i > 0, the degenerated sequence
∗
yields isomorphisms
H0(Y,Rjp F) ∼= Hj(X,F).
∗
leray 1.2.1. On an affine scheme or variety X any coherent sheaf has vanishing cohomology
Hi(X,F) = 0 for i > 0. Therefore, for an affine open covering U = (U ) of a scheme X
α
the conditions for the Leray isomorphisms Hi(U,F) ∼= Hi(X,F) between Cˇech– and functorial
cohomology are fulfilled. This gives a convenient tool to compute the cohomology on algebraic
schemes or varieties once their gluing data are known. The easiest non–trivial example is the
projective space. Here we have the invertible sheaves O(m) given by the cocycle (g ) with
ij
respect to the covering U = {x 6= 0} with g = (x /x )m. The sheaf O(m) is also the sheaf
i i ij j i
5
R(m)∼ where R = k[x ,...,x ] and R(m) = R . Using Cˇech–cohomology one can prove,
0 n i i+m
Ha
see [14], III, 5.1,
1.2.2. Lemma: Hi(P (k),O(m)) = 0 for 0 < i < n.
n
We are going to use Cˇech–cohomology also in the proof of the
kuenn 1.2.3. Lemma: Let S and X be varieties and let p and q be the first and second projection of
S × X. Then for any coherent module F on X the canonical homomorphisms
k
O ⊗Hj(X,F) → Rjp q∗F
S ∗
are isomorphism for all j.
Proof. The tensor product on the left is defined as the sheaf U 7→ O (U)⊗ Hj(X,F) and the
S k
homomorphism by the cup product H0(U,O )⊗Hj(X,F) → Hj(U ×X,q∗F) followed by the
S
natural map Hj(U ×X,q∗F) = Hj(p−1U,q∗F) → H0(U,Rjp q∗F). If U ⊂ S and V ⊂ X are
∗
both affine open subsets and U = SpecA,V = SpecB, there is a finitely generated B–module
F with F|V ∼= F. Then q∗(F|V) on U ×V is given by q∗F = (A⊗ F)∼ and we have
k
Γ(U,O )⊗ Γ(V,F) ∼= A⊗ F ∼= Γ((A⊗ F)∼) ∼= Γ(U ×V,q∗F).
e S k k k e
Now let V = (V ) be an open affine covering of X and U ⊂ S as before. By the definition of
α
ˇ
the Cech–groups and the last chain of isomorphism we get isomorphisms
O (U)⊗Cj(V,F) −→≈ Cj(U ×V,q∗F)
S
and from this that the mappings
O (U)⊗Hj(X,F) → Hj(U ×X,q∗F)
S
(cid:3)
are isomorphisms. Performing direct limits at each side proves the Lemma.
f
projmor 1.3. Projective morphisms: A morphism X −→ Y of varieties or schemes over k is called
projective if it can be factored locally over Y into a closed immersion X ⊂ Y ×P (k) and the
n
Ha EGA II
projection onto Y [14]. The precise definition of [4] is that there is a coherent sheaf E on Y
such that X can be immersed into the projective bundle P(E) of E, and f is induced by the
projection of P(E) onto Y. The variety P(E) will be discussed in section 7.
p
cohdir 1.3.1. Proposition: Let X −→ S be a projective morphism and F a coherent module on X.
Then Rjp F is coherent for any j.
∗
EGA III
Remark1: Grothendieck proved in[5], 3.2that moregenerally Rjp F is coherent forany proper
∗
morphism. In particular, in the absolute case where S is a point, the groups Hi(X,F) of a
coherent sheaf on a complete variety X over k are finite dimensional k–vector spaces.
Remark 2: If S is affine, Hi(S,Rjp F) = 0 for i > 0 and we have isomorphisms
∗
H0(S,Rjp F) ∼= Hj(X,F),
∗
such that Hj(X,F) is a finitely generated Γ(S,O )–module and such that
S
Rjp F = Hj(X,F)∼.
∗
6
Proof. The proof will be done in the category (Sch/k). Since the statement is local with respect
to S we may assume thatS is affine, X ⊂ S× P where P = P , andwe may even assume that
k n,k
X is equal to the product by extending F trivially. Now S = SpecA with A = k[y ,...,y ]/a
1 m
and
S × P = Proj A[x ,...,x ],
k 0 n
Ha
and F = F for a graded finitely generated A[x ,...,x ]–module F, see [14], II, 5.15. Let
0 n
Ha
R = A[x ,...,x ]. By Serre’s theorem, see [14], II, 5.17, there is a ”free” module L = R[−m]k,
0 n
e
where R[−m] = R , and a surjection L → F → 0. Applying the functor ∼ we obtain an
i i−m
exact sequence
0 → G → L → F → 0
with L = L = O ⊠O (−m)k. Note here that O (−m) is the invertible sheaf on P obtained
S P P
from k[x ,...,x ](−m). Since P can be covered by n + 1 open affine sets, we have Hj(U ×
0 n
e
P,F) = 0 for j > n and U ⊂ S affine open. We get the exact sequence
0 → Rn−1p F → Rnp G → Rnp L → Rnp F → 0,
∗ ∗ ∗ ∗
kuenn
where Rnp L ∼= O ⊗Hn(P,O (−m))k by 1.2.3. Because Hj(P,O (−m)) = 0 for 0 < j < n,
∗ S P P
we have 0 at the left hand side. Now we can apply the same to G and can represent Rnp G as a
∗
quotient of Rnp L′ for some L′ of the same type as L. Then Rnp F is coherent as the cokernel
∗ ∗
of coherent sheaves, and the same is true for Rnp G. It follows that also Rn−1p F is coherent.
∗ ∗
From this it is clear that descending induction applies to show that Rjp F is coherent for all
∗
j ≤ n. (cid:3)
f
fibrest 1.4. Restriction to fibres: Let X −→ Y be a morphism of varieties over k and let F,G be
modules on X,Y respectively. For any y ∈ Y the fibre X is defined and its ideal sheaf is the
y
image of f∗I in O , where I is the ideal sheaf of y in Y. We are given the pull-back
{y} X {y}
diagram
(cid:31)(cid:127)
X // X
y
iy
fy f
(cid:15)(cid:15) (cid:31)(cid:127) (cid:15)(cid:15)
{y} // Y
jy
We write G(y) for j∗G ∼= G /m G and F for i∗F ∼= F ⊗ O . By the commutativity of the
y y y y y y OX Xy
diagram we get a canonical isomorphism,
(f∗G) ∼= O ⊗ G(y).
Xy Xy k
There is also a canonical homomorphism
(f F)(y) → H0(X ,F ).
∗ y y
This is induced by
(f F) = limH0(f−1U,F) → H0(X ,F )
∗ y y y
−−→
U∋y
7
because m (f F) is mapped to 0. We also obtain the diagram
y ∗ y
O ⊗ (f F)(y)
Xy k ∗
(cid:15)(cid:15) NNNNNNαN(NyN)NNNN''
O ⊗H0(X ,F ) // F
Xy y y ev y
where ev is the evaluation map of F and α(y) is the composition. It is an exercise in the
y
language of sheaves to show that α(y) is nothing but the restriction to the fibre of the canonical
homomorphism f∗f F → F modulo the isomorphism f∗(f F) ∼= O ⊗(f F)(y).
∗ ∗ y Xy ∗
1.4.1. Corollary: If f∗f F → F is surjective, then for any y ∈ Y the sheaf F is generated
∗ y
by global sections.
Proof. If f∗f F → F is surjective, then its restriction to a fibre is surjective, i.e. α(y) is
∗
surjective, and then ev is surjective. (cid:3)
banda 1.5. Theorem: (Serre, Grothendieck). Let S be a variety over k and P = P (k), let F be a
n
coherent sheaf on S ×P and let p and q be the first and second projection. Let F(m) denote
F ⊗ q∗O (m). Then for any affine open subset S ⊂ S there is an integer m such that for
P 0 0
m ≥ m
0
(A) p∗p F(m) → F(m) is surjective over S
∗ 0
(B) Rjp F(m) = 0 for j > 0 over S .
∗ 0
Ha
Proof. see [14], II, 5.17. The proof is a corollary of the proof of the coherence of the Rjp F.
∗
cohdir
We may assume that S is affine. With the notation of 1.3.1 there is an exact sequence
0 → G → L → F → 0
onS×P withL = O ⊠Ok(−d). Ifm−d ≥ −nwehaveRnp L(m) = O ⊗Hn(P,Ok(m−d)) =
S P ∗ S P
0 and hence Rnp F(m) = 0. With m eventually enlarged, also Rnp G(m) = 0 and then
∗ ∗
Rn−1p F(m) = 0. By induction we may assume that Rip F(m) = 0 for m ≥ m , and also
∗ ∗ 0
Rip G(m) = 0 for m ≥ m . This proves (B). Since R1p G(m) = 0 we obtain the diagram
∗ 0 ∗
p∗p L(m) // p∗p F(m) // 0
∗ ∗
(cid:15)(cid:15) (cid:15)(cid:15)
L(m) // F(m) // 0
But for m−d ≥ 0 the left vertical arrow is surjective because this is the homomorphism
O ⊠O ⊗H0(P,Ok(m−d)) → O ⊠Ok(m−d) .
S P P S P
induced from the evaluation map of Ok(m−d) on P. This proves (A). (cid:3)
P
Remark: In the absolute case with S a simple point, the two statements are the classical
theorems (A) and (B): If F is a coherent sheaf on P, there exists an integer m such that for
0
m ≥ m
0
(A) the natural evaluation map Γ(P,F(m))⊗O → F(m) is surjective
P
8
(B) Hj(P,F(m)) = 0 for j > 0.
9
2. Flatness and Base Change
For coherent sheaves F on P (k) the Euler characteristic
n
χF = (−1)ν dimHν(P (k),F)
n
ν≥0
X
Ha
is an important invariant, see [14]. It is proved there that the function m 7→ χF(m), where
F(m) = F ⊗ OPn(k)(m), is a numerical polynomial. This polynomial is called the Hilbert
polynomial. Inthissection werecall brieflythebehaviour ofthepolynomialχF (m) forfamilies
s
(F ) of coherent sheaves on P (k) together with base change properties for direct images.
s s∈S n
bch 2.1. Base change: Given a Cartesian or pull–back diagram of varieties
g
Y // X
q p
(cid:15)(cid:15) (cid:15)(cid:15)
T // S
f
and a sheaf of O –modules on X, there are canonical homomorphisms
X
f∗Rjp F → Rjq g∗F
∗ ∗
for any j ≥ 0, called base change homomorphisms. They can be defined as follows. For any
open subset U ⊂ X there are canonical homomorphisms
Hj(U,F) → Hj(g−1U,g•F) → Hj(g−1U,g∗F),
where g•F is the topological pull back and where the second homomorphism is induced by
g•F → g•F ⊗g•OX OY = g∗F, whileEGtAheIIIfirst homomorphism is derived from Γ(U,−) →
Γ(g−1U,g•−) via derived functors, see [5], Ch 0, 12.1. Using this, we get for an open set U ⊂ S
the composition
Hj(p−1U,F) // Hj(g−1p−1U,g∗F)
Hj(q−1f−1U,g∗F) // H0(f−1U,Rjq g∗F).
∗
This defines a sheaf homomorphism
Rjp F → f Rjq g∗F
∗ ∗ ∗
which corresponds to homomorphisms above. When T is a point s ∈ S with inclusion f, then
the base change homomorphism reduces to a homomorphism
(Rjp F)(s) −ϕ−j−(→s) Hj(X ,F ).
∗ s s
fibrest
This generalizes the homomorphism α(s) considered in 1.4 in case j = 0. The proof of the
Mu
following proposition can be found in [28], p. 50, (i).
10
bchzero 2.2. Proposition: Let S be an algebraic variety over k and P = P (k), and let F be a coherent
n
f
sheaf on S×P. Then for any morphism T −→ S there is an integer m such that for any m ≥ m
0 0
the base change homomorphism
f∗p F(m) −→≈ q g∗F(m)
∗ ∗
is an isomorphism. In particular, for any s ∈ S there is an integer m such that for m ≥ m
s s
the homomorphism
p (F(m))(s) → H0(P ,F (m))
∗ s s
is an isomorphism, P = {s}×P.
s
flss 2.3. Flatness: The same statement for the higher direct images is in general only true under
the additional assumption that F be flat over S. Let us recall the flatness condition here, see
p
Appendix A. If X −→ S is any morphism of varieties and M an O –module, M is said to be
X
flat over S at x ∈ X if the module M is a flat module over O under O → O . M
x S,p(x) S,p(x) X,x
is said to be S–flat or flat over S if it is flat over S at any point. We say that X is flat over S
if O is S–flat. The following criterion is proved in commutative algebra, see Appendix A.
X
M is flat over S iff for any s ∈ S the sheaf T orOX(M,O ) = 0, because the stalk of this sheaf
1 Xs
at any x ∈ X is isomorphic to the module TorOX,x(M ,O /m O ).
s 1 x X,x S,s X,x
The most important use of flatness is made for restriction to fibres. Let 0 → M′ → M →
M′′ → 0 be an exact sequence of O –modules and let M′′ be S–flat. Then for any point s ∈ S
X
the restricted sequence
0 → M′ → M → M′′ → 0
s s s
flfibre
is exact on the fibre X , see Appendix A 17.6.1. More generally, pullbacks of flat sheaves are
s
flbch
flat, see 17.6. In particular:
When X × Y −→p X is a product projection and M any O –module, then p∗M is Y–flat on
k X
the product.
The following theorem is indispensable for dealing with direct images. Its proof may be found
Mu Ha
in EGA III, 7.7 or in [28], p. 51 or in [14], III, 12.11.
p
bchthm 2.4. Theorem: Let X −→ S be a projective morphism and let F be a coherent and S–flat
module on X, and let
(Rip F)(s) −ϕ−i−(→s) Hi(X ,F )
∗ s s
be the base change homomorphisms for points. Then
(1) If ϕi(s ) is surjective, there is a neighbourhood U(s ) in S such that for any morphism
0 0
f
T −→ U the base change homomorphism
f∗Rip F −→≈ Riq g∗F
∗ ∗
bch
is an isomorphism (notation as in 2.1).
(2) If ϕi(s ) is surjective, then ϕi−1(s ) is surjective if and only if Rip F is free in a neigh-
0 0 ∗
bourhood U of s .
0