Invariant Hilbert schemes and desingularizations of quotients by classical groups

0.4 MB·English

by  Ronan Terpereau

INVARIANT HILBERT SCHEMES AND DESINGULARIZATIONS
OF QUOTIENTS BY CLASSICAL GROUPS
3
1 RONANTERPEREAU
0
2
Abstract. LetW beafinite-dimensionalrepresentation ofareductivealge-
n braicgroupG. TheinvariantHilbertschemeHisamodulispacethatclassifies
a the G-stable closed subschemes Z of W such that the affine algebra k[Z] is
J
thedirectsumofsimpleG-moduleswithprescribed multiplicities. Inthisar-

7 ticle, we consider the case where G is a classical group acting on a classical
1 representation W and k[Z] is isomorphic to the regular representation of G
as aG-module. Weobtain familiesofexamples where His asmoothvariety,

] andthus forwhichtheHilbert-Chowmorphism γ∶ H→W//Gisacanonical
G
desingularization ofthecategorical quotient.
A
.
h
t
a Contents
m
1. Introduction and statement of the main results 1
[
2. Generalities on invariant Hilbert schemes 6

1 3. Case of SL acting on (kn)⊕n′ 9
v n
4. Case of GL acting on (k2)⊕n1⊕(k2∗)⊕n2 14
0 2
2 References 30
0
4
.
1
1. Introduction and statement of the main results
0
3
Themainmotivationforthisarticlecomesfromaclassicalconstructionofcanon-
1
icaldesingularizationsofquotientvarieties. Letk beanalgebraicallyclosedfieldof
:
v characteristiczero,Gareductivealgebraicgroupoverk,andW afinite-dimensional
i
X linear representation of G. We denote
r ν ∶ W →W//G
a
the quotient morphism, where W//G ∶= Spec(k W G is the categorical quotient.
[ ] )
In general, ν is not flat and the variety W G is singular. A universal "flattening"
//
of ν is given by the invariant Hilbert scheme constructed by Alexeev and Brion
([AB05, Bri]). We recall briefly the definition (see Section 2 for details). Let
Irr G be the set of isomorphism classes of irreducible representations of G and h
a f(unc)tion from Irr G to N. Such a function h is called a Hilbert function. The
invariant Hilbert sc(hem) e HilbG W parametrizes the G-stable closed subschemes
h
( )
Z of W such that
k Z ≅ M⊕h(M)
[ ] ⊕
M∈Irr(G)
as G-module. If h = h is the Hilbert function of the generic fiber of ν, then we
W
denote
∶=HilbG W
H hW( )
1
2 RONANTERPEREAU
to simplify the notation. Let ⊂ ×W, equipped with the first projection π ∶
X H
→ , be the universal family over . The morphism π is flat by definition and
X H H
we have a commutative diagram
π
//
X H
p γ
(cid:15)(cid:15) (cid:15)(cid:15)
W //W G
ν //
where p is the second projection and γ is the Hilbert-Chow morphism that sends a
closed subscheme Z ⊂W to the point Z G⊂W G. The Hilbert-Chow morphism
is projective and induces an isomorphism// over th//e largest open subset W G ⊂
∗
W G over which ν is flat. In particular, ν is flat if and only if ≅ W( G//. )The
// H //
main component of is the subvariety defined by
H
main∶=γ−1 W G .
∗
H (( // ) )
Then the restriction
γ ∶ main→W G
H //
is a projective birational morphism.
Question. In which cases is the Hilbert-Chow morphism, possibly restricted to the
main component, a desingularization of W G?
//
When G is a finite group, we recall some relevant results:
● If dim W =2, then is always a smooth variety. In particular, if W =k2
and if(G⊂)SL W , tHhen Ito and Nakamura showed that γ is the minimal
( )
desingularization of the quotient surface W G (see [IN96, IN99]).
● If dim W =3 and G⊂SL W , Bridgeland/, King and Reid showed, using
( ) ( )
homological methods, that once again is a smooth variety and that γ is
H
a crepant desingularization of W G (see [BKR01]).
● If dim W = 4 and G ⊂ SL W ,/then can be singular. For instance, if
G⊂SL( i)s the binary tetrah(edr)al groupHand if W is the direct sum of two
2
copies of the defining representation,then Lehn and Sorgershowedthat
has two irreducible components but that main is smooth (see [LS]). H
H
However, when G is infinite, this question is open and completely unexplored. In
this article, we study the case where G is a classical group and W is a classical
representation of G. We then show that the invariant Hilbert scheme is still a
H
desingularizationofW Gin"small"cases(butnotingeneral). Inaddition,unlike
//
the case where G is finite, it may happen that W G is smooth but ν is not flat;
//
then γ is not an isomorphism.
LetV,V′,V andV bevectorspacesofdimensionn,n′,n andn respectively.
1 2 1 2
We consider the following cases:
(1) G=SL V actingnaturallyonW ∶=Hom V′,V =V⊕n′,the directsumof
n′ copie(s o)f the defining representation; ( )
(2) G=O V acting naturally on W ∶=Hom V′,V =V⊕n′;
(3) G=Sp( V) , with n even, acting naturally(on W)∶=Hom V′,V =V⊕n′;
(4) G =GL( V) acting naturally on W ∶= Hom V ,V ⊕Ho(m V,V) = V⊕n1 ⊕
1 2
V∗⊕n2, (the)direct sum of n copies of the(definin)g repres(entat)ion and n
1 2
copies of its dual.
INVARIANT HILBERT SCHEMES AND DESINGULARIZATIONS OF QUOTIENTS 3
In these four cases, the description of the quotient morphism ν is well-known
and follows from First Fundamental Theorem for the classical groups ([Pro07,
§9.1.4,§11.1.2,§11.2.1]):
● Case 1. ν is the natural map
Hom V′,V → Hom Λn V′ ,Λn V ≅ Λn V′∗ ,
(w ) ↦ ( Λ(n w) ( )) ( )
( )
where Λn V′∗ denotes the n-th exterior power of V′∗. We distinguish be-
( )
tween three different cases:
-if n′<n, then W G= 0 and ν is trivial;
-if n′=n, then W//G=Λ{n}V′∗ ≅A1 and ν w =det w ;
-if n′ >n, then W//G=C (Gr n),V′∗k is the(a)ffine co(ne)over the Grass-
mannian Gr n,V′∗ //of the(n-d(imensio)n)al subspaces of V′∗ viewed as a
subvariety of(P ΛnV)′∗ via the Plücker embedding.
One may check(that W) G=Λn V′∗ if and only if n=1 or n≥n′−1. We
will see in Section 3 tha/t/ν is fla(t if a)nd only if n=1 or n′≤n.
● Case 2. ν is the composite map
Hom V′,V → Hom S2 V′ ,S2 V → S2 V′∗ ,
(w ) ↦ ( S(2 w) ( )) ↦ q S(2 w)
1
( ) ( ( ))
whereS2 V′∗ denotesthesymmetricsquareofV′∗,andq isthemorphism
1
induced b(y th)e linear projection from S2 V onto the line generated by a
( )
non-degenerate quadratic form. It follows that
W G=S2 V′∗ ≤n∶= Q∈S2 V′∗ rk Q ≤n
// ( ) { ( ) ∣ ( ) }
is a symmetric determinantal variety. One may check (see [Ter, Corollaire
3.1.7]) that ν is flat if and only if n≥2n′−1.
● Case 3. ν is the composite map
Hom V′,V → Hom Λ2 V′ ,Λ2 V → Λ2 V′∗ ,
(w ) ↦ ( Λ(2 w) ( )) ↦ q Λ(2 w)
2
( ) ( ( ))
whereq isthemorphisminducedbythelinearprojectionfromΛ2 V onto
2
( )
the line generated by a non-degenerate skew-symmetric bilinear form. It
follows that
W G=Λ2 V′∗ ≤n∶= Q∈Λ2 V′∗ rk Q ≤n
// ( ) { ( ) ∣ ( ) }
isaskew-symmetricdeterminantalvariety. Onemaycheck(see[Ter,Corol-
laire 3.3.8]) that ν is flat if and only if n≥2n′−2.
● Case 4. ν is the natural map
Hom V ,V ×Hom V,V → Hom V ,V ,
1 2 1 2
( ) ( ) ( )
u ,u ↦ u ○u
1 2 2 1
( )
and thus
W G=Hom V ,V ≤n∶= f ∈Hom V ,V rk f ≤n
1 2 1 2
// ( ) { ( ) ∣ ( ) }
isadeterminantalvariety. We willseeinSection4thatν isflatifandonly
if n≥n +n −1.
1 2
The main result of this article is the following
4 RONANTERPEREAU
Theorem. In the following cases, the invariant Hilbert scheme is a smooth
H
variety and the Hilbert-Chow morphism is the succession of blows-up described as
follows:
● Case 1. Let C be the blow-up of the affine cone C Gr n,V′∗ at 0.
0
If n′ >n>1, then is isomorphic to C . ( ( ))
0
● Case 2. Let s bHe the blow-up of the symmetric determinantal variety
0
S2 V′∗ ≤n at 0Y.
–( If)n′>n=1 or n′=n=2, then is isomorphic to s.
0
– Ifn′ >n=2, then is isomorphHic totheblow-upofYs alongthestrict
0
transform of S2 VH′∗ ≤1. Y
● Case3. Let a be the(blow)-up of the skew-symmetric determinantal variety
0
Λ2 V′∗ ≤n aYt 0.
–( If )n′=n=4, then is isomorphic to a.
0
– If n′ > n = 4, thenH is isomorphic Yto the blow-up of a along the
0
strict transform of ΛH2 V′∗ ≤2. Y
● Case 4. Let be the blow(-up o)f the determinantal variety Hom V ,V ≤n
0 1 2
Y ( )
at 0.
– If max n ,n >n=1 or n =n =n=2, then is isomorphic to .
1 2 1 2 0
– If min(n ,n )≥ n = 2 and max n ,n > 2, thHen is isomorphicYto
1 2 1 2
the blo(w-up o)f along the stric(t trans)form of HomH V ,V ≤1.
0 1 2
Y ( )
When W G is singular and Gorenstein, we will see that the desingularization
//
γ is never crepant. Also, we conjecture that in Cases 1-4, the invariant Hilbert
scheme issmoothifandonlyifν isflatorweareinoneofthecasesofthe above
H
theorem. In this direction, we will show in another article (also partially extracted
from[Ter]) that is singularinCases2and4forn=3,andalsointhe casewhere
G=SO V actsHnaturally on W =Hom V′,V with n′=n=3.
( ) ( )
A key ingredient in the proof of the main Theorem is a group action on
H
with finitely many orbits. Indeed, for any reductive algebraic group G, any finite
dimensional G-module W, and any algebraic subgroup
G′⊂AutG W ,
( )
it is known that G′ acts on W Gand , and that the quotient morphismand the
Hilbert-Chow morphism are G//′-equivaHriant. To describe the flat locus of ν, it is
almostenoughto knowthe dimensionofthe fiber of ν overone pointofeachorbit.
Inthesameway,determiningthetangentspaceof atapointofeachclosedorbit
H
is enough to show that is smooth, thanks to a semicontinuity argument.
H
Another important ingredient of this article, which was already used by Becker
in [Bec, §4.1], is the
Key-Proposition. Let G, W and G′ be as above. For any M ∈ Irr G , there
exists a finite-dimensional G′-submodule F ⊂ HomG M,k W that(ge)nerates
M
HomG M,k W ask W G,G′-module,andthereexistsa(G′-e[quiv])ariantmorphism
( [ ]) [ ]
δ ∶ →Gr h M ,F∗ .
M W M
H ( ( ) )
Thanksto the Key-Proposition,we obtainthe followingresultwhichshowsthat
it suffices to describe in "small" cases to understand all cases:
H
Reduction Principle. Let G and W be as in Cases 1-4. We suppose that n′ ≥n
resp. n ,n ≥n, and we fix E ∈Gr n,V′∗ resp. E ,E ∈Gr n,V∗ ×Gr n,V .
1 2 1 2 1 2
( ) ( ) ( ) ( )
INVARIANT HILBERT SCHEMES AND DESINGULARIZATIONS OF QUOTIENTS 5
In Cases 1-3, we denote W′ ∶= Hom V′ E⊥,V resp. in Case 4, we denote W′ ∶=
Hom V E⊥,V ×Hom V,E , where(E⊥/resp. )E⊥, denotes the orthogonal subspace
1 1 2 1
to E(in /V′ resp). to E (in V ).
1 1
Then the invariant Hilbert scheme is the total space of a homogeneous bundle
over Gr n,V′∗ in Cases 1-3 resp. Hover Gr n,V∗ ×Gr n,V in Case 4, whose
1 2
fiber is i(somorp)hic to ′∶=HilbG W′ . ( ) ( )
H hW′( )
Forinstance,totreatthecaseofGL actingonV⊕n1⊕V∗⊕n2 withn ,n ≥2,we
2 1 2
just have to consider V⊕2⊕V∗⊕2. The reduction principle is the most important
theoretical result of this article and will certainly be helpful to determine further
examples of invariant Hilbert schemes.
To show the main Theorem, we have to proceed case by case but we follow
a general method: First, we perform the reduction step. Then, we look for the
closed G′-orbits in , where G′ is a reductive algebraic subgroup of AutG W . In
H ( )
Cases 1-4, such orbits are projective and thus contain fixed-points for the action
of a Borel subgroup B′ ⊂G′. To determine these fixed-points, we significantly use
representationtheoryofG′. InCases1-4,weshow that hasonly one fixed-point
H
that we denote Z . We deduce from Lemma 2.6 that is connected and that Z
0 0
belongs to the main component main. We then detHermine the Zariski tangent
spaceT andwecheckthatitsHdimensionisthe same asthatof main. We thus
get thatZ0H= main is a smooth variety. H
H H
It is known that there exists a finite subset of Irr G such that the morphism
E ( )
γ× δ ∶ Ð→W G× Gr h M ,F∗
M W M
∏ H // ∏ ( ( ) )
M∈E M∈E
is a closed embedding; this is a consequence of the construction of the invariant
HilbertschemeasaclosedsubschemeofthemultigradedHilbertschemeofHaiman
and Sturmfels ([HS04]). This suggests to choose an appropriate simple representa-
tion M ∈Irr G and to check whether γ×δ is a closed embedding of . If this
1 ( ) M1 H
holds,thenwehavetoidentifytheimage;otherwise,wechooseanothersimplerep-
resentationM andwe look if γ×δ ×δ is a closedembedding. This procedure
2 M1 M2
must stop after a finite number of steps and we get an explicit closed embedding
of as simple as possible.
H
In Section 2, we recall some basic results and we give a proof of the Key-
Proposition. Case 1 (the easiest one) is treated in Section 3. Case 4 (the most
difficult one) is treated for n=2 in Section 4. The details for the other cases (ex-
cept Case 2 for n=1 which is an easy exercice left to the reader!) can be found in
the thesis [Ter] from which this article is extracted.
To conclude, let us mention that we can also use invariant Hilbert schemes to
constructcanonicaldesingularizationsofsomesymplecticvarieties. LetG⊂GL V
beasinCases2-4andW =V⊕n′⊕V∗⊕n′. ThenW isasymplecticrepresentatio(no)f
G and one can define a moment map µ∶ W →g∗, where g is the Lie algebra of G.
Thesymplecticreduction ofW isdefinedasµ−1 0 G. Wewillseeinaforthcoming
article (also extracted from [Ter]) that, in "sm(all)"//case, γ ∶ main →µ−1 0 G is
H ( )//
a desingularization,sometimes symplectic, but that is reducible in general.
H
Acknowledgments: I am very thankful to Michel Brion for proposing this sub-
ject to me and for a lot of helpful discussions, I thank Hanspeter Kraft for ideas
6 RONANTERPEREAU
and corrections concerning Section 4.3, and I thank Tanja Becker for very helpful
discussions during her stay in Grenoble in October 2010.
2. Generalities on invariant Hilbert schemes
2.1. Thesurvey[Bri]givesadetailedintroductiontotheinvariantHilbertschemes.
In this section, we recall some definitions and useful properties of these schemes.
All the schemes we consider are supposed to be separated and of finite type over
k. Let G be a reductive algebraic group and N a rational G-module, we have the
decomposition
N ≅ N ⊗M,
(M)
⊕
M∈Irr(G)
where N ∶= HomG M,N is the space of G-equivariant morphisms from M to
(M)
( )
N. The G-module N ⊗M is called the isotypic component of N associated to
(M)
N and dim N is the multiplicity of M in N. If, for any M ∈Irr G , we have
(M)
dim N (<∞, t)hen we define ( )
(M)
( )
h ∶ Irr G → N
( )
M ↦ dim N
(M)
( )
the Hilbert function of N.
Let S be a scheme, a G-scheme and π ∶ → S an affine morphism, of finite
type and G-invariant. ZAccording to [Bri, §2.3Z], the sheaf ∶= π admits the
∗ Z
F O
following decomposition as a (sheaf of) ,G-modules:
S
O
(1) ≅ ⊗M.
(M)
F ⊕ F
M∈Irr(G)
The action of G on comes from the action of G on each M, and each ∶=
(M)
F F
HomG M, is a coherent G-module. From now on, we suppose that the family
π is m(ultipFlic)ity finite, thatFis to say, G is a coherent -module. If, in addition,
S
F O
π is flat, then each -module is locally free of finite rank, and this rank is
S (M)
constant over each cOonnected coFmponent of S. Let h ∶ Irr G → N be a Hilbert
( )
function and W a finite-dimensional G-module.
Definition 2.1. We define the Hilbert functor HilbG W : Schop→Sets by
h
( )
(cid:31)(cid:127)
//S×W is a G-stable closed subscheme;
S ↦⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ Z π ## S(cid:15)(cid:15)p1 RRRRRRRRRRRRRRRRRR FππZ∗MiOsZias≅flloa⊕ctaMmlly∈oIrrfrpr(ehGei)soFmfM;ra⊗nMk ;h(M) over OS. ⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭.
An element π ∶ → S ∈ HilbG W S is called a flat family of G-stable
h
closed subschem(es of WZ over)S. By [Br(i, T)h(eo)rem 2.11], the functor HilbG W is
h
represented by a quasi-projective scheme HilbG W : the invariant Hilbert s(che)me
h
( )
associated to the G-module W and to the Hilbert function h. We recall that we
denote ∶= HilbG W , where h is the Hilbert function of the generic fiber of
ν ∶ WH→ W G.hWW(e )denote ⊂WW × , equipped with the second projection
// X H
π∶ → , the universal family over .
X H H
INVARIANT HILBERT SCHEMES AND DESINGULARIZATIONS OF QUOTIENTS 7
Proposition 2.2. [Bri, Proposition 3.15] With the above notation, the diagram
q
(2) // W
X
ν
π
(cid:15)(cid:15) (cid:15)(cid:15)(cid:15)(cid:15)
//W G
H γ //
commutes, where π and q are the natural projections.
Moreover, the pull-back of γ to the flat locus of π is an isomorphism.
We fix an algebraic subgroup G′ ⊂AutG W . Then we have:
( )
Proposition 2.3. [Bri, Proposition 3.10] With the above notation, G′ acts on
and on such that all the morphisms of Diagram (2) are G′-equivariant. H
X
Remark 2.4. With the above notation, the morphism π# ∶ → ∶= π is a
H ∗ X
morphism of G′-modules, and the that appear in theOdecompFositionO(1) are
(M)
,G′-modules. F
S
O
We are now ready to show the Key-Proposition stated in the introduction:
Proof of the Key-Proposition. We use the notation of Diagram (2). The inclusion
ι ∶ ⊂ W × is G′ ×G-equivariant and so ι induces a surjective mor-
W//G
phismXof ,G′ ×GH-modules p → ∶= π . But p =
OH 2∗OH×W//GW F ∗OX 2∗OH×W//GW
⊗ k W , where we recall that k W G =k W G. We can then consider
H k[W//G]
Othe decompos[itio]n into ,G′×G-module[s // ] [ ]
H
O
⊗ k W ≅ ⊗ k W ⊗M,
OH k[W//G] [ ] ⊕ OH k[W//G] [ ](M)
M∈Irr(G)
where the action of G′ on ⊗ k W is induced by the action of G′ on W,
H k[W//G]
andG′ actstriviallyonM.OForeachM ∈[ Ir]r G ,wededuceasurjectivemorphism
of ,G′-modules ( )
H
O
(3) ⊗ k W ↠ .
OH k[W//G] [ ](M) F(M)
Itfollowsthatthevectorspacek W generates =HomG M, as ,G′-
[ ](M) F(M) ( F) OH
module. The vectorspace k W is generallyinfinite-dimensional,but k W
[ ](M) [ ](M)
is a k W G-module of finite type, and thus there exists a finite-dimensional G′-
modul[e F] that generates k W as k W G-module:
M [ ](M) [ ]
(4) k W G⊗F ↠k W .
[ ] M [ ](M)
We deduce from (3) and (4) a surjective morphism of ,G′-modules
H
O
(5) ⊗F ↠ ,
H M (M)
O F
where we recallthat is a locally free -module of rankh M . By [EH01,
(M) H W
F O ( )
Exercice 6.18], such a morphism gives a morphism of schemes
δ ∶ →Gr dim F −h M ,F ,
M M W M
H ( ( ) ( ) )
and one may check that δ is G′-equivariant. Finally, we identify
Gr dim F −h M ,F ≅Gr h M ,F∗ ,
M W M W M
( ( ) ( ) ) ( ( ) )
which completes the proof. (cid:3)
8 RONANTERPEREAU
Remark 2.5. The Key-Proposition holds more generally if we consider a Hilbert
function h such that h V =1, where V denotes the trivial representation.
0 0
( )
We now obtain a set-theoretic description of the morphism δ . We recall that,
M
for any G-module M, we have the canonical isomorphisms
k W ∶= HomG M,k W ≅ M∗⊗k W G ≅ MorG W,M∗ .
(6) [ ](M) ( [ ]) ( [ ]) ( )
m↦φ m f φ⊗f w↦f w φ
( ( ) ) ( ( ) )
Via these isomorphisms, the elements of the G′-module F ⊂ k W identify
M (M)
with G′equivariant morphisms from W to M∗. The map δ is giv[en]by:
M
δ ∶ →Gr dim F −h M ,F , Z ↦Ker f ,
M M W M Z
H ( ( ) ( ) ) ( )
where
f ∶ F ↠
(7) Z M FM,Z
q ↦ q
∣Z
is the surjective linear map obtained by passing to the fibers in (5).
2.2. WefixaBorelsubgroupB′⊂G′. Wewillobtainaseriesofelementaryresults
that will be useful in the next sections to show that is a smooth variety in some
H
cases.
Lemma 2.6. We suppose that W G has a unique closed G′-orbit and that this
orbit is a point x. Then, each G′-/s/table closed subset of contains at least one
fixed-point for the action of the Borel subgroup B′. MoreoHver, if has a unique
H
fixed-point, then is connected.
H
Proof. The Hilbert-Chow morphism γ is projective and G′-equivariant, so the set-
theoreticfiberγ−1 x isaprojectiveG′-variety. LetC beaclosedsubsetof ,then
γ C is a G′-stab(le)closed subset of W G. Hence x ∈ γ C , that is, C ∩Hγ−1 x
( ) // ( ) ( )
is non-empty. Therefore, the Borel fixed-point Theorem ([Bor91, Theorem 10.4])
yields that C ∩γ−1 x contains at least one fixed-point for the action of B′. As
( )
a consequence, each connected component of contains at least one fixed-point,
hence the last assertion of the lemma. H (cid:3)
Lemma 2.7. We suppose, as in Lemma 2.6, that W G has a unique closed orbit
x for the action of G′, and we denote B′ the set/o/f fixed-points for the Borel
subgroup B′. Then we have the equivalenHce
= main is a smooth variety ∀Z ∈HB′, dim(TZH)=dim(Hmain); and
H H ⇔{ is connected.
H
Proof. The direction is easy. Let us prove the other implication. We denote
d∶= dim main . The⇒set E ∶= Z ∈ k dim T >d is a G′-stable closed
Z
subsetof(H k .)IfE isnon-empt{y,thHen(E)co∣ntain(sonHefi)xed-}pointofB′ byLemma
2.6. LetZH(be)thisfixed-point,thenwehavedim T >d,whichcontradictsour
0 ( Z0H)
assumption. It followsthat E is empty, andthus is a smoothvariety. Since is
connected by assumption, has to be irreducibleH, and thus = main. H(cid:3)
H H H
Let Z ∈ be a closed point, I ⊂k W the ideal of Z viewed as a subscheme of
Z
W, and R∶H=k W I the algebra of g[lob]al sections of the structure sheaf of Z.
Z
[ ]/
INVARIANT HILBERT SCHEMES AND DESINGULARIZATIONS OF QUOTIENTS 9
Proposition2.8. [Bri,Proposition3.5]With the above notation, thereis a canon-
ical isomorphism
T ≅HomG I I2,R ,
Z R Z Z
H ( / )
where HomG stands for the space of R-linear, G-equivariant maps.
R
Next, let N be a G-submodule of k W contained in I such that the natural
Z
morphism of R,G-modules δ ∶ R⊗N[→]I I2 is surjective; and let R be a G-
Z/ Z
submodule of R⊗N such that we have the exact sequence of R,G-modules
R⊗R Ðρ→ R⊗N Ðδ→ I I2 →0,
(8) Z/ Z
f ⊗1 ↦ f
where we denote f the image of f ∈I in I I2.
Z Z/ Z
Applying the leftexactcontravariantfunctor Hom .,R to the exactsequence
R
( )
(8) and taking the G-invariants, we get the exact sequence of finite-dimensional
vector spaces
(9) 0 // HomG I I2,R δ∗ //HomG R⊗N,R ρ∗ //HomG R⊗R,R
R( Z/ Z ) R( ) R( )
≅ ≅
(cid:15)(cid:15) (cid:15)(cid:15)
HomG N,R HomG R,R
( ) ( )
Therefore, we have T H ≅ Im δ∗ = Ker ρ∗ . Moreover, if the ideal I is B′-
Z Z
stable,thenwecanchooseN and(Ra)sB′×G(-m)odulessuchthatallthemorphisms
oftheexactsequence(8)aremorphismsofR,B′×G-modulesandallthemorphisms
of the exact sequence (9) are morphisms of B′-modules.
Lemma 2.9. With the above notation, suppose that R≅k G as G-module. Then,
dim HomG I I2,R =dim N −rk ρ∗ . [ ]
In p(articuRla(r,Zi/f δZis a))n isomo(rph)ism,(the)n dim HomG I I2,R =dim N .
( R( Z/ Z )) ( )
Proof. We have
dim HomG I I2,R =dim HomG R⊗N,R −rk ρ∗
R Z Z R
( ( / )) ( ( )) ( )
=dim HomG N,R −rk ρ∗
( ( )) ( )
=dim MorG G,N∗ −rk ρ∗ since R≅k G ,
( ( )) ( ) [ ]
=dim N −rk ρ∗ .
( ) ( )
(cid:3)
3. Case of SL acting on kn ⊕n′
n
( )
3.1. We denote G∶=SL V , G′ ∶=GL V′ , and W ∶=Hom V′,V . Consider the
action of G′×G on W giv(en)by: ( ) ( )
(10) ∀w∈W, ∀ g′,g ∈G′×G, g′,g .w∶=g○w○g′−1.
( ) ( )
We recall that W G was described in the introduction. We also recall that
C Gr n,V′∗ denote//s the affine cone over Gr n,V′∗ , and that C denotes the
0
bl(ow-u(pofC))Gr n,V′∗ at0. Theaimofthisse(ctioni)stoshowthemainTheorem
( ( ))
in Case 1. Specifically, we will show:
10 RONANTERPEREAU
Theorem 3.1. If n=1 or n′ ≤n, then H≅W G and the Hilbert-Chow morphism
//
γ is an isomorphism.
If n′>n>1, then H≅C and γ is the blow-up of C Gr n,V′∗ at 0.
0
( ( ))
In all cases, H is a smooth variety and thus, when W G is singular, γ is a desin-
//
gularization.
The cases n = 1 or n′ ≤ n are easy and are treated by Corollary 3.4. The case
n′>n>1 is handled by Proposition 3.10.
3.2. Generic fiber and flat locus of the quotient morphism. One cancheck
that the variety W G is smooth except when 1 < n < n′−1 in which case W G
// //
has a unique singularity at 0. Moreover, W G is always normal ([SB00, §3.2,
//
Théorème 2]) and Gorenstein ([SB00, §4.4, Théorème 4]) because G is semisimple.
When n′ ≥ n, the variety W G is the union of two G′-orbits: the origin and its
//
complement, denoted U.
Proposition 3.2. Let U be the open orbit defined above.
1) If n′≥n, then the fiber of ν over a point of U is isomorphic to G.
2) If n′>n>1, then U is the flat locus of ν; otherwise, ν is flat.
Proof. The first assertion is a well-known fact and is easily checked. Let us show
the second assertion. If n′ < n, then ν is trivial, hence ν is flat. If n′ = n, then
W G≅A1 and ν is the determinant, which is flat by [Har77, Exercice 10.9].
We//now sukppose that n′>n. We know that ν is flat overa non-empty open subset
of W G ([Eis95, Theorem14.4]), hence ν is flat over U by G′-homogeneity. Then,
one c/a/n check that dim ν−1 0 =dim w∈W rk w ≤n−1 = n′+1 n−1 .
By 1), the dimension of(the(fib))er of ν(o{ver a po∣int(of)U is n2}−)1. (The fib)(ers of)a
flat morphism have all the same dimension; hence, if n>1, U is the flat locus of ν.
Otherwise, W G is smooth and all the fibers of ν have the same dimension, and
thus ν is flat b/y/ [Har77, Exercice 10.9]. (cid:3)
Corollary 3.3. If n′≥n, the Hilbert function h of the generic fiber of ν is given
W
by:
∀M ∈Irr G , h M =dim M .
W
( ) ( ) ( )
We deduce from Propositions 2.2 and 3.2:
Corollary 3.4. The Hilbert-Chow morphism is an isomorphism if and only if:
● n′<n, and then H is a reduced point; or
● n′=n, then H≅A1 and det∶ W →A1 is the universal family; or
k k
● n′>n=1, then H≅V′∗ and Id∶ V′∗→V′∗ is the universal family.
It remains to consider the case n′ >n>1. To do this, we will use the reduction
principle for SL that will allow us to reduce to n′=n.
n
3.3. Reduction principle for SL . From now on, we suppose that n′ ≥ n. We
n
are going to show in Case 1 the reduction principle stated in the introduction.
Specifically, we will prove:
Proposition 3.5. We suppose that n′ ≥ n and let P be the parabolic subgroup of
G′ = GL V′ that preserves a n-dimensional subspace of V′∗. Then we have a
G′-equiva(rian)t isomorphism
ψ ∶ G′×PA1 ≅ H,
k
g′,x P ↦ g′.x
( )