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INVARIANT DEFORMATIONS OF ORBIT CLOSURES IN sl (n )S′e bastien Jansou and Nicolas Ressayre Abstract We study deformations of orbit closures for the action of a con-nected semisimple group G on its Lie algebra g ,especially when G is the special linear group.The tools we use are on the one hand the invariant Hilbert scheme and on the other hand the sheets of g .We show that when G is the special linear group,the connected components of the invariant Hilbert schemes we get are the geometric quotients of the sheets of g .These quotients were constructed by Katsylo for a general semisimple Lie algebra g ;in our case,they happen to be a?ne spaces.Introduction Let G be a complex reductive group,and V be a ?nite dimensional G -module.A fundamental problem is to endow some sets of orbits of G in V with a structure of variety.The geometric invariant theory is the classical answer in this context:the set of closed orbits of G in V has a natural structure of a?ne variety.We denote by V //G this variety,equipped with a G -invariant quotient map π:V →V //G .Recently,Alexeev and Brion de?ned in [AB]a structure of quasiprojec-tive scheme on some sets of G -stable closed a?ne subscheme of V .A natural
question is to wonder what happens when one applies Alexeev-Brion’s con-struction to the orbit closures of G in V .Here,we study this construction in the case of a well known G -module,namely the adjoint representation of a semisimple group G ,especially when G is the special linear group SL(n ).From now on,we assume that G is semisimple,and denote by g its Lie algebra endowed with the adjoint action of G .Let us recall that a sheet of g is an irreducible component of the set of points in g whose G -orbit has a ?xed dimension.Let us ?x a sheet S .We show that the G -module structure on the a?ne algebra C [G ·x of x doesn’t depend on x in S .This allows us to de?ne a set-theoretical application from S to some Alexeev-Brion’s invariant Hilbert scheme of g :
πS :S ?→Hilb G S (g )
x ?→
A unique sheet is open in g:we call it the regular one,and denote it by g reg.
In Section2we are interested in Hilb G g
reg (g).The graph of the quotient
mapπ:g→g//G is a?at family of G-stable closed subschemes of g over g//G.So,this family is the pullback of the universal one by a morphism.
We prove that this morphism is an isomorphism by showing that Hilb G g
reg (g)
is smooth and applying Zariski’s main theorem.So,we obtain that the applicationπg
reg
identi?es with the restriction of the quotient mapπ:g→g//G;in particular,it is a morphism.
In Section3,we study any sheet S for G=SL(n).We explicitly con-struct a?at family over an a?ne space whose?bers are the closures in g of the G-orbits in S.Then,we show following the same method as in the case of g reg that this family is universal.Let us denote byπ:S→S/SL(n) the geometric quotient of S,constructed by Katsylo in[Ka].We show that there is a canonical morphism
θ:S/SL(n)?→Hilb SL(n)
S (g)
SL(n)·x?→
weightλ).
Let us recall some de?nitions from[AB,§1].A family of a?ne G-schemes over some scheme S is a scheme X equipped with an action of G and with a morphismπ:X→S that is a?ne,of?nite type and G-invariant.We have a G-equivariant morphism of O S-modules
π?O X?
Fλ?C V(λ),
λ∈Λ+
is equipped with the trivial action of G.Let where each Fλ:=(π?O X)U
(λ)
h:Λ+→N be a function.The family X is said to be of Hilbert function h if each Fλis an O S-module locally free of rank h(λ).(Then the morphism πis?at.)
Let X be an a?ne G-scheme,and h:Λ+?→N a function.A family of G-stable closed subschemes of X over some scheme S is a G-stable closed subscheme X?S×X.The projection S×X→S induces a family of a?ne G-schemes X→S.The contravariant functor:(Schemes)??→(Sets) that associates to every scheme S the set of families X?S×X of Hilbert function h is represented by a quasiprojective scheme denoted by Hilb G h(X) ([AB,§1.2].
The dimension of an a?ne G-scheme whose a?ne algebra has?nite mul-tiplicities can be read on its Hilbert function:
Proposition1.1.Let h:Λ+?→N be a function.Let Y and Z be two a?ne schemes of Hilbert function h.Then dim Y=dim Z.
Proof.Let us denote by A the a?ne ring of Y.
If Y is horospherical,that is([AB,Lemma2.4])if for any dominant weightsλ,μ,we have A(λ)·A(μ)?A(λ+μ),it is clear that the dimension of Y can be read on its Hilbert function.Indeed,let us denote byθ0the linear map fromΛ?Q to Q which associates to any fundamental weight the value1.We denote byθthe group homomorphism fromΛto Z that is the restriction ofθ0.We associate toθa graduation of the algebra A by N:its homogeneous component of degree d is
A(λ).
A d:=
λ∈Λ+,θ(λ)=d
The dimension of A d is?nite,and depends only on h:
dim A d=
h(λ)dim V(λ).
λ∈Λ+,θ(λ)=d
So the Hilbert polynomial of the graded algebra A depends only on h,and so does the dimension of Y.
We can deduce the proposition.Indeed,Y admits a?at degeneration over a connected scheme to a horospherical G-scheme Y′that admits the same Hilbert function(by[AB,Theorem2.7]).So dim Y=dim Y′depends only on h.
We will use the method of“asymptotic cones”of Borho and Kraft([PV,§5.2]):let V be a?nite dimensional rational G-module and F the closure of an orbit in V(or,more generally,any G-stable closed subvariety contained in a?ber of the categorical quotient V→V//G).We embed V into the projective space P(C⊕V)of vector lines of C⊕V by the inclusion v→[1⊕v]. The closure of F in P(C⊕V)is denoted by
F is the closed cone X generated by F.
The vector space C⊕V,equipped with its natural scheme structure,is denoted by A1×V.The cone X?A1×V,viewed as a reduced closed sub-scheme,is a?at family of a?ne G-schemes.Its?bers over non-zero elements are homothetic to F.Its?ber over0is a reduced cone,denoted by?F.It is contained in the null-cone of V(that is the?ber of the categorical quotient V→V//G containing0).Its dimension is the same as F.
We consider the adjoint action of G on its Lie algebra g.If x is an element of g,the a?ne algebra of the closure of its orbit,viewed as a reduced scheme, has?nite multiplicities.Let us denote by h x its Hilbert function;we call it the Hilbert function associated to x.In this paper,we are interested in the
connected component denoted Hilb G x of the scheme Hilb G h
x (g)that contains
G·x embedded in g.We determine it when x is in g reg in§2,and for any x when G is the special linear group in§3.
Let us denote by G x the stabilizer of x in G,and g x its Lie algebra.The coadjoint action of G x is its natural action on the dual vector space g?x. Proposition1.2.Let us assume the orbit closure
G·x
Hilb G x to Hilb G x at the point
G·x at the point x is g.x;it is stable under the action of G x.We denote by[g/g.x]G x the space of invariants under the action of G x on the quotient vector space g/g.x.According to[AB, Proposition1.15(iii)],we have a canonical isomorphism
T
G·x is assumed to be normal.Moreover,every orbit in g has even dimension,and has a?nite number of orbits in its closure ([PV,Corollary3page198]),so the codimension of the boundary of G·x in
To transform(1)into the isomorphism of the proposition,we will use the Killing form on g,denoted byκ.As g is semisimple,its Killing form gives an isomorphism
φ:g?→g?
y?→κ(y,·).
The isomorphismφis G-equivariant,thus G x-equivariant.It sends g.x onto the space g⊥x of linear forms on g that vanish on g x.Indeed,the common zeros of the elements ofφ(g.x)are the elements y in g such that
?z∈g,κ([z,x],y)=0,
that is
?z∈g,κ(z,[x,y])=0,
and this last condition means that y belongs to g x sinceκis non-degenerate.
Thus the short exact sequence of G x-modules
0?→g.x?→g?→g/g.x?→0
identi?es(thanks toφ)with
0?→g⊥x?→g??→(g x)??→0,
and the proposition follows from(1).
A sheet of g is a maximal irreducible subset of g consisting of G-orbits of a?xed dimension.Every sheet of g contains a unique nilpotent orbit.A regular element of g is an element of g whose orbit has maximal dimension. The open subset of g whose elements are the regular elements is a sheet denoted by g reg.
Let us call Hilbert’s sheet a maximal irreducible subset of g consisting of elements admitting a?xed associated Hilbert function.
Proposition1.3.The Hilbert’s sheets of g coincide with its sheets. Proof.According to Proposition1.1,any Hilbert’s sheet is contained in some sheet.It just remains to check that two points of some sheet S have the same associated Hilbert function.
Let F be the closure of an orbit in S.We recalled that its asymptotic cone?F is a degeneration of F.In particular,it is contained in the closure of S.Moreover,?F is contained in the null-cone of g,and its dimension is the same as F.So?F is the closure of the nilpotent orbit of S.
The a?ne algebra of g is the symmetric algebra of g?.Its graduation induces a G-invariant?ltration on the a?ne algebra A of F.The a?ne algebra of the asymptotic cone?F is isomorphic,as an algebra equipped with an action of G,to the graded algebra?A associated to the?ltered algebra A.In particular,A and?A are isomorphic as G-modules,and their multiplicities are equal:the Hilbert function of F is equal to that of?F,and the proposition is proved.
Notice that in the case of the regular sheet,Proposition1.3is a direct consequence of[Ko,Theorem0.9].
2Regular case
Let us denote by h reg the Hilbert function associated to the regular elements of g(Proposition1.3).In this section,we prove that the invariant Hilbert
scheme H reg:=Hilb G h
reg (g)is the categorical quotient g//G,that is an a?ne
space whose dimension is the rank of G.
By[Ko,Theorem0.1],all schematic?bers of the quotient morphism g→g//G are reduced.This allows us to identify in the following the schematic?bers with the set-theoretical?bers.
2.1A morphism from g//G to H reg
Let X reg be the graph of the canonical projection g→g//G.It is a family of G-stable closed subschemes of g over g//G.
Proposition2.1.The closed subscheme X reg is a family of G-stable closed subschemes of g with Hilbert function h reg.
Proof.Let us denote byπ:X reg→g//G the canonical projection,and by R:=π?O X
reg
the direct image byπof the structural sheaf of X reg.We have
to prove that for any dominant weightλ,we have that R U
(λ)is a locally free
sheaf on g//G of rank h(λ).
Let us?rst study the case whereλ=0.The morphismπ//G:X reg//G→
g//G induced byπis clearly an isomorphism.So R G=R U
(0)is a free module
on g//G of rank1=h reg(0).
Letλbe a dominant weight.It is known(see[AB,Lemma1.2])that
R U
(λ)is a coherent R G-module.Thus it is a coherent module on g//G.To
see that it is locally free,we just have to check that its rank is constant.The ?bers ofπare those of the canonical projection g→g//G,so they are the orbit closures of the regular elements,and all of them admit h reg as Hilbert
function.So the rank of R U
(λ)at any closed point of g//G is h(λ),and the
proposition is proved.
This gives us a canonical morphism
φreg:g//G?→H reg.
We will prove in the following of§2thatφreg is an isomorphism.
Lemma2.2.The morphismφreg realizes a bijection from the set of closed points of g//G to the set of closed points of H reg.
Proof.We remark thatφreg is injective.Let us check it is surjective:in other words,that any G-invariant closed subscheme of g of Hilbert function h reg is a?ber of g→g//G.
Let Y be such a subscheme.As h reg(0)=1,it has to be contained in some?ber F of g→g//G over a reduced closed point.But F already corresponds to a closed point of H reg in the image ofφreg.Moreover,F admits no proper closed subscheme admitting the same Hilbert function,so F=Y,and the lemma is proved.
Let us denote by r the rank of G.The quotient g//G is an a?ne space of dimension r.A consequence of Lemma2.2is:
Corollary2.3.The dimension of H reg is r.
2.2Tangent space
In this section,we prove:
Proposition2.4.The scheme H reg is smooth.
Proof.Let Z be a closed point of H reg.We have to prove that the dimension of the tangent space T Z H reg is r.We still denote by Z the closed subscheme of g corresponding to Z.By Lemma2.2,we know that Z is a?ber of the morphism g→g//G,thus the closure of some regular element x.It is a normal variety.By Proposition1.2,we have to prove that the dimension of
(g?x)G x
is r,or simply that it is lower or equal to r(by Corollary2.3).
Let us prove that the dimension of the bigger space
(g?x)g x
is r,and the proposition will be proved.
A linear form on g x is g x-invariant i?it vanishes on the derived algebra [g x,g x],so we have to prove that
(g x/[g x,g x])?
is r-dimensional.We will prove that g x is an r-dimensional abelian algebra, and the proposition will be proved.This is true if x is semisimple,because then g x is a Cartan subalgebra of g.If the regular element x is not assumed to be semisimple,the dimension of g x is still r,because this doesn’t depend on the regular element x,by de?nition.Let us check that g x is abelian.
Let us denote by Grass r(g)the grassmannian of r-dimensional subspaces of g,endowed with its projective variety structure.The subset of g reg×Grass r(g):
{(z,h)∈g reg×Grass r(g)|h·z=0and[h,h]=0}
is closed,so its image by the natural projection into g reg is closed too.As its image contains the semisimple elements of g reg,it is equal to g reg.Thus
g x is abelian for any regular x,and the proposition is proved.
2.3Conclusion
We can now conclude that the family X reg of Proposition2.1is the universal family:
Theorem2.5.The morphismφreg from g//G to H reg is an isomorphism. Proof.The morphismφreg is bijective(Lemma2.2)and H reg is normal. According to Zariski’s main theorem,φreg is an isomorphism.
Remark2.6.One knows there is a canonical morphism
ψreg:H reg?→g//G
that associates to any closed point F of H reg(viewed as a closed subscheme of g)its categorical quotient F//G(viewed as a closed point of g//G).This morphism is a particular case of morphism
η:Hilb G h(V)?→Hilb h(0)(V//G)
de?ned in[AB,§1.2],because h reg(0)=1and thus the punctual Hilbert scheme that parametrizes closed subschemes of length1in g//G identi?es with g//G itself.The morphismψreg is clearly the inverse morphism ofφreg. Remark2.7.As pointed to us by M.Brion,Theorem2.5admits the fol-lowing generalization:
Let X be an irreducible a?ne G-variety such thatπ:X→X//G is ?at.Let h be the Hilbert function of its?bers.Then the graphΓofπis the universal family;in particular,Hilb G h(X)identi?es with X//G.
The idea of his proof is to check thatΓrepresents the functor.Let X?X×S be a?at family of Hilbert function h,over some a?ne scheme S.Since h(0)=1,the scheme S identi?es with X//G and maps on X//G (by the morphism induced by the?rst projection X×S→X).We obtain the following commutative diagram:
X
It remains to prove that X is isomorphic(canonically)to the?ber prod-uctΓ×X//G S.This has only to be veri?ed over the closed points of S.The assertion follows.
The Hilbert schemes we obtain applying the above Brion’s result to G-modules are always a?ne spaces.The representations V of a simple group G such that V→V//G is?at have been classi?ed by G.Schwarz in[Sch].
Unfortunately,the sheets of sl(n)are not a?ne in general and Katsylo’s quotient cannot be extended to their closure.So,Brion’s theorem cannot be applied,whereas the method we used to prove Theorem2.5can be used. 3Case of sl(n)
We denote by t an indeterminate over C,and I n the identity matrix of size n×n.If x is an element of sl(n)and i=1···n,we denote by Q x i(t)the monic greatest common divisor(in the ring C[t])of the(n+1?i)×(n+1?i)-sized minors of x?tI n,and Q x n+1(t):=1.
Then we put
q x i(t):=Q x i(t)/Q x i+1(t).
The polynomials q x1(t),···,q x n(t)are the invariant factors of the matrix x?tI n with coe?cients in the euclidean ring C[t],ordered in such a way that q x i+1(t)divides q x i(t).
If x,y are elements of sl(n),then y is in the closure of the orbit SL(n)·x of x if and only if for any i=1...n,the polynomial Q x i(t)divides Q y i(t). In other words,i?for any i,the polynomial Q x i(t)divides the(n+1?i)×(n+1?i)-sized minors of y?tI n.
According to[W],when x is nilpotent,these conditions de?nes the clo-sure of SL(n)·x as a reduced scheme:to be more precise,when one divides a (n+1?i)×(n+1?i)-sized minor of y?tI n by Q x i(t)using Euclid algorithm, the remainder he gets is a regular function of y.All such functions generate the ideal of the closure of SL(n)·x.We will deduce easily from this di?cult result that the same remains true if x is no longer assumed to be nilpotent.
The set of sheets of sl(n)is in bijection with the set of partitions n,that is of sequencesσ=(b1≥b2≥b3≥...)of nonnegative integers such that b1+b2+b3+···=n(see[Bo,§2.3]).Namely,ifσis a partition of n,the elements of the correspondent sheet Sσare those elements x such that for any i,the polynomial q x i(t)is of degree b i.We denote by σ=(c1≥c2≥c3≥...) the conjugate partition,where c j is the number of i such that b i≥j.We denote by hσthe Hilbert function associated to the points of Sσ(Proposition 1.3).We denote by Zσthe closure of the nilpotent orbit of Sσ.The connected
component of Hilb SL(n)
hσ(sl(n))that contains Zσas a closed point is denoted
Hσ.We will prove in this section that Hσis an a?ne space of dimension b1?1.The proof is similar to§2.
We recall that the sheets of sl(n)are smooth([Kr]).
3.1A construction of the geometric quotient of Sσ
Katsylo showed in[Ka]that any sheet of a semisimple Lie algebra admits a geometric quotient.Although his proof contains an explicit construction,it doesn’t make clear the geometric properties of the quotient.Here we present a simple description of the quotient in the case of the Lie algebra sl(n).It takes on the invariant factors theory.We get that the quotient is an a?ne space.
Lemma3.1.Given some i,the application Sσ?→A b i that associates to any x the coe?cients of q x i(t)=t b i+λx b
i?1
t b i?1+···+λx0t0is regular. Proof.Up to scalar multiplication,the polynomial q x i(t)is the unique nonzero polynomial of degree less or equal to b i such that
dim ker q x i(x)≥N:=
b i
j=1
c j.(2)
Thus the closed subset of Sσ×P b i consisting of elements(x,[μ0:···:μb
i ])
such that
dim ker(b i
j=0
μj x j)≥N
is the graph of the application
ψ:Sσ?→P b i
x?→[λx0:···:λx b
i?1
:1]
According to[Hr,Exercise7.8p76],this graph is also the graph of a rational mapφfrom Sσto P b i.On the open subset?of Sσwhereφis regular,φcoincides withψ,so the functions x→λx j are regular functions from?to A1.As Sσis smooth,the complementary of?in Sσhas codimension at least2([Sha,Thm3chap II.3.1]).We conclude that the functions extend to regular functions from Sσto A1.By continuity,these extensions satisfy (2),so they coincide with the functions x→λx j on Sσ.
Let us de?ne,for any x in Sσ,the monic polynomial of degree b i?b i+1:
p x i(t):=q x i(t)/q x i+1(t)
(where q x n+1:=1).It follows from the previous lemma that its coe?cients, viewed as functions of x,are regular functions from Sσto A1.
Given an x,the family(p x1(t),...,p x n(t))can be any family of monic polynomials of degrees b i?b i+1,provided the following relation is satis?ed, where S(p x i)denotes the sum of the roots of p x i,counted with multiplicities (given by its?rst nondominant coe?cient):
n
iS(p x i)=0
i=1
(this relation simply means that the trace of x is zero).
Thus,associating to any x the coe?cients of the family(p x1(t),...,p x n(t)), we get a regular mapπfrom Sσto a linear hyperplane of C b1,which we will denote by A b1?1.
Proposition3.2.The mapπ:Sσ?→A b1?1is the geometric quotient of Sσ.
Proof.This map is surjective,and its?bers are exactly the orbits of Sσunder the action of SL(n).Let us denote by Sσ/SL(n)the geometric quotient of Sσ(whose existence is proved in[Ka]).The mapπis the composite of the canonical projection from Sσto Sσ/SL(n)with a regular bijection
Sσ/SL(n)?→A b1?1.
This last map is bijective(thus birational),and the space A b1?1is normal. According to Zariski’s main theorem,it is an isomorphism.
3.2A morphism from Sσ/SL(n)to Hσ
If z=(p1(t),...,p n(t))is a closed point of A b1?1corresponding to the orbit SL(n)·x in Sσ,the polynomial
Q x i(t)=p i(t)·(p i+1(t))2·...·(p n(t))n?i+1
only depends on z.Let us denote it by Q z i(t).Its coe?cients are regular functions from A b1?1to A1.
Let us consider the closed subscheme Xσof{(z,y)∈A b1?1×sl(n)} de?ned by the vanishing,for i=1...n,of the remainders we get when we divide the(n+1?i)×(n+1?i)-minors of y?tI n by Q z i(t).We denote by Iσthe ideal generated by these remainders.The underlying set of Xσconsists of all the couples(z,y)such that y is in the closure of the orbit corresponding to z.
Proposition 3.3.The closed subscheme Xσis a family of SL(n)-stable closed subschemes of sl(n)with Hilbert function hσ.
Proof.The proof is similar to that of Proposition2.1.The subscheme Xσis a family of SL(n)-stable closed subschemes of sl(n)over A b1?1.Let us denote byπthe morphism Xσ?→A b1?1.
As previously,let us?rst remark that the morphism
π//SL(n):Xσ//SL(n)?→A b1?1
induced byπis an isomorphism.To do this,let us verify that the comor-phism
(π//SL(n))?:C[A b1?1]?→C[A b1?1]?C[sl(n)]SL(n)/I SL(n)
σ
is an isomorphism.It is injective,asπis surjective.Its surjectivity comes from the relations that de?ne Xσ:they give,for i=1,that Q z1(t)divides the determinant of tI n?y,that is the characteristic polynomial of y.As their degrees are equal,Q z1(t)and the characteristic polynomial of y are equal. This gives the surjectivity.
We go on as previously:letλbe a dominant weight.The R SL(n)-module
R U (λ)is of?nite type([AB,Lemma1.2]).Thus(π?O X
σ
)U
(λ)
is a coherent
O A b
1?1-module.To see that it is locally free,we just have to check that its
rank is constant.Let us assume that the origin0∈A b1?1corresponds to the nilpotent orbit in Sσ.The?ber ofπover0is the closure of this orbit,
?tted with its structure of reduced scheme.Thus,the rank of(π?O X
σ)U (λ)
at0is hσ(λ).If z is any point of A b1?1,the?ber ofπover z is as a set the closure in sl(n)of the corresponding orbit.So,by Proposition1.3the
rank of(π?O X
σ)U
(λ)
at z is at least hσ(λ).To conclude,we use the action of
the multiplicative group on sl(n)(by homotheties)and the induced action on A b1?1,that makesπequivariant.The orbit of z goes arbitrary close to 0,and the rank of a coherent sheaf is upper semicontinuous,so the rank of
(π?O X
σ)U
(λ)
is hσ(λ)at z.
3.3Tangent space
In this section,we compute the dimension of the tangent space to Hσat the point Zσ:
Proposition3.4.The dimension of T Z
σ
Hσis b1?1.
Proof.Let x be an element in the open orbit in Zσ.It is known that Zσis normal([KP]).So by Proposition1.2,we just have to prove that the dimension of
(sl(n)?x)SL(n)x
is b1?1.Let us consider SL(n)as a closed subgroup of the general linear group GL(n),and sl(n)as a subalgebra of gl(n).The stabilizer GL(n)x of
x in GL(n)is generated by SL(n)x and the center of GL(n).It is clearly equivalent to prove that the dimension of
(gl(n)?x)GL(n)x
is b1.The group GL(n)x is connected,so the last space is isomorphic to
(gl(n)?x)gl(n)x.
A linear form on gl(n)x is gl(n)x-invariant i?it vanishes on the derived algebra[gl(n)x,gl(n)x],so we have to prove that
(gl(n)x/[gl(n)x,gl(n)x])?
is b1-dimensional.This fact is the following elementary lemma. Lemma 3.5.Let E= c1i=1E i be a graded vector space over C,where each E i is b i-dimensional.We denote by h:=gl(E)the Lie algebra of endomorphisms of E.Let x be a nilpotent element of h such that each subspace E i is stabilized by x,and the restriction of x to each E i is cyclic.
Let us denote by h x the stabilizer of x in h.Then the codimension of the derived algebra[h x,h x]in h x is b1.
Proof.The graduation of E induces a graduation on the vector space h:
h= i,j Hom(E i,E j).
Let us denote by p i:E?→E i the natural projections.As they commute with x,the subspace h x of h is homogeneous:
h x= i,j Hom x(E i,E j),
where Hom x(E i,E j)denotes the space of homomorphisms that commute with x.Let us choose,for any i,an element e i of E i such that x b i?1e i=0. We put n ij:=b j?b i if j
Hom x(E i,E j)=C[x]·f ij.
We notice that if i=j,then Hom x(E i,E j)is contained in[h x,h x]. Indeed,for any u:E i→E j,we have[u,p i]=u.
So we have to prove that the codimension in i Hom x(E i,E i)of
[h x,h x]∩ i Hom(E i,E i)
is b1.The last vector space is generated by its elements of the form
P(x)[f ji,f ij]=P(x)x|b i?b j|(id E
i ?id E
j
),
where P(x)is a polynomial in x.
One checks easily that a basis of a supplementary in i Hom x(E i,E i) of this space is given by the family of elements
x k id E
i
where0≤k
3.4Conclusion
In this section,we prove that the family Xσof Proposition3.3is the universal family:
Theorem3.6.The morphismφσfrom Sσ/SL(n)to Hσobtained in§3.2is an isomorphism.
We denote by
Sσ)which parametrizes the closed subschemes of
Sσwith Hilbert func-tion hσis the closure of some orbit in Sσ.
Let X be such a subscheme.As hσ(0)=1,it has to be contained in some ?ber F of the categorical quotient Sσ//SL(n)over a reduced closed point.But F already corresponds to a closed point of H′σin the image of ψσ.Moreover,F admits no proper closed subscheme admitting the same Hilbert function,so F=X,and the lemma is proved.
Corollary3.8.The dimension of H′σis b1?1.
The action of the multiplicative group G m on sl(n)by homotheties in-duces canonically an action of G m on Hσ,and on H′σ(because it stabilizes
Proposition3.9.Let F be a closed point of H′σ.The morphismη:G m?→H′σ,t?→t.X extends to a morphism A1?→H′σ,0?→Zσ.
Proof.The point F corresponds to a SL(n)-invariant closed subscheme of
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锈铁钉与稀盐酸、稀硫酸反应 一、实验目的： 1、探究稀盐酸、稀硫酸的化学性质。 2、归纳酸的通性。 二、实验原理： 1、酸能使指示剂变色。 2、金属活动性顺序表中，氢前面的金属能把稀盐酸、稀硫酸中的氢置换出来。 3、稀盐酸、稀硫酸能与铁锈反应。 三、实验用品： 点滴板、胶头滴管、试管8支、镁条、锌粒、铁钉、稀盐酸、稀硫酸、石蕊、酚酞溶液、锈铁钉 四、实验过程： 1、用胶头滴管在点滴板指定的凹穴内分别各滴 2～3滴稀盐酸、稀硫酸，再向其中分别滴1滴石蕊、 酚酞溶液，并观察现象。 2、回忆第八单元所学的几种金属分别与稀盐酸或 稀硫酸的反应，写出化学方程式。 讨论：上面反应的生成物有什么共同之处？ 3、在盛有稀盐酸和稀硫酸的试管里分别放入一根 生锈（铁锈的主要成分是Fe2O3）的铁钉，过一会儿取出铁钉，用水洗净，铁钉表面和溶液颜色有什么变化？ 讨论： ①上面反应的生成物有什么共同之处？ ②利用上面的反应可以清除铁制品表面的锈，除锈时能否将铁制品长时间浸在酸中？为什么？ ③根据以上实验和讨论，试归纳出盐酸、硫酸等酸有哪些相似的化学性质。 结论： 1、酸能与指示剂作用 酸溶液能使石蕊溶液变红色，使酚酞溶液不变色。 2、酸能与活泼金属反应：活泼金属 + 酸 = 盐 + 氢气 Mg+2HCl=MgCl2+H2↑ Zn+2HCl=ZnCl2+H2↑ Fe+2HCl=FeCl2+H2↑ Mg+H2SO4=MgSO4+H2↑ Zn+H2SO4=ZnSO4+H2↑ Fe+H2SO4=FeSO4+H2↑ 3、酸能与某些金属氧化物反应：碱性氧化物（某些金属氧化物）+ 酸 = 盐 + 水 Fe2O3+6HCl=2FeCl3+3H2O Fe2O3+3H2SO4=Fe（SO4）3+3H2O 五、实验说明： 1、用点滴板既方便又节约药品，但滴加的量不能过多。 2、绣铁钉要浸入稀盐酸和稀硫酸中2～3分钟即可。 3、用1：1的稀盐酸和1：4的稀硫酸，效果最好。
重要的酸 -----区别稀盐酸和稀硫酸 临山镇中王品 一教学目标 1、科学探究:引导学生利用所学知识设计探究实验,讨论优化实验方案并予以真实呈 现,使学生有充分机会体验探究过程. 2、科学知识与技能: 能区分稀盐酸和稀硫酸，知道氯化银和硫酸钡是不溶于稀硝酸 的白色沉淀。在演示实验中观察和了解胶头滴管滴加液体的正 确操作，观察和记录实验现象 3、情感、态度与价值观: 让学生在教学和实验探究中认识到科学必须尊重事实,能 根据客观事实提出自己的想法和建议,重视分工合作意识 和创新意识的培养.增加学生对科学的求知欲和对探究的 兴趣. 4、科学、技术与社会的关系:初步认识科学技术的发展推动社会生产力的进步,科学 技术也深入到我们生产和生活的各个方面. 二教学重点与难点 教学重点: 了解区分稀硫酸和稀盐酸的原理和方法 教学难点: 培养学生对实验步骤、现象分析和对突发事件的处理；从离子的角度认识稀硫酸和稀盐酸的检验。 三教学准备 碳酸钠（Na2C03）、稀盐酸、稀硫酸、硝酸银(AgNO3)、紫色石蕊试液、氯化钡(BaCl2)、稀硝酸、试管、胶头滴管、废液槽 学生课前准备：复习酸的化学性质探究实验的结论,了解酸跟盐反应的规律和条件 要求. 五教学过程 一、创设情境 今天我们要来学习的是如何区别稀盐酸和稀硫酸。在重要的酸第一课时中，我们认识了浓盐酸和浓硫酸，也详细了解了其各自的物理性质特点，也能根据这些特点将两者区分开来。但大家看看我今天带来的这两瓶稀盐酸和稀硫酸你能想办法区分二者吗? 展示两种稀酸溶液并让学生从已知的知识角度去总结其性质. 问题1:两种酸在颜色上有差异吗? 生:没有,都是无色溶液 问题2:稀盐酸还有浓盐酸的挥发性吗?
课题名称：金属的化学性质课题编号：初中24号 《金属与盐酸、稀硫酸的反应》微课教学设计 清远市清新区禾云镇第一初级中学陈金炜 教学目标 1．知识与技能 初步认识常见金属与盐酸、硫酸的置换反应，能用置换反应解释一些与日常生活有关的化学问题。 2．过程与方法 认识科学探究的基本过程，能进行初步的探究活动；初步学会运用比较、归纳、概括等方法对获取的信息进行加工，使学生逐步形成良好学习习惯和方法。 3．情感态度与价值观 培养学生勤于思考、勇于创新实践、严谨求实的科学精神。 导入 很多金属不仅能与氧气反应，而且还能与盐酸或稀硫酸反应。金属与盐酸或稀硫酸能否反应以及反应的剧烈程度，也可反映金属的活泼程度。下面，我们就通过实验来比较镁、锌、铁、铜的活动性。 探究活动 金属与盐酸、稀硫酸的反应 1、观看实验视频； 金属 现象反应的化学方程式 稀盐酸稀硫酸稀盐酸稀硫酸 镁 锌 铁 铜 3、讨论： 哪些金属能与盐酸、稀硫酸发生反应反应的剧烈程度如何反应后生成什么气体哪些金属不能与盐酸、稀硫酸发生反应根据反应时是否有氢气产生，将金属分为两类。 对于能发生的反应，从反应物和生成物的物质类别如单质、化合物的角度分析，这些反应有什么特点将这一类反应与化合反应、分解反应进行比较。 教师点拨
1、这几个反应的共同特点为：由一种单质与一种化合物反应，生成另一种单质和另一种化合物。 2、概念：由一种单质与一种化合物反应，生成另一种单质和另一种化合物的反应叫做置换反应。 3、表示式：A+BC==B+AC 4、结论：镁、锌、铁的金属活动性比铜的强，它们能置换出盐酸或稀硫酸中的氢。 5、根据金属能否与盐酸、稀硫酸反应及其反应的剧烈程度可以判断金属活动性的强弱。 知识应用 1、根据上述探究实验的现象，你能推断出：镁、锌、铁、铜中， 的化学性质最活泼， 的化学性质最不活泼；活动性由强到弱的顺序为： 。 2、判断下列反应的类型： 2Mg+O 2 2MgO 3Fe+2O 2 Fe 3O 4 ○3Fe+H 2SO 4==FeSO 4+H 2↑ ○42Al+3CuO 4=Al(SO 4)3+3Cu ○ 52LiOH+CO 2═Li 2CO 3+H 2O ○62KMnO 4=====K 2MnO 4+MnO 2+O 2↑ 应用拓展 1、将X 、Y 、Z 三块大小相同的金属片分别投入到10﹪的稀盐酸中，X 表面无明显现象，Y 表面缓慢地产生气泡，Z 表面迅速产生大量气泡，则X 、Y 、Z 的金属活动顺序为（ ） A 、X>Z>Y B 、Z>Y>X C 、X>Y>Z D 、Z>X>Y 2、右图是甲、乙、丙、丁四种常见金属与盐酸反应的比较示意图，上述金属可分为两类，则丙应和_______分为一类，若乙为金属锌，则丙可能是____________。 本课小结 很多金属不仅能与氧气反应，而且还能与盐酸或稀硫酸反应。金属与盐酸或稀硫酸能否反应以及反应的剧烈程度，也可反映金属的活动性：能与盐酸、稀硫酸发生的比不反应的强；反应越剧烈的活动性越强。 △ 甲乙丙丁 气泡
課題《居家化學實驗探究》 由澳門科學技術發展基金資助 鋁和稀鹽酸,稀硫酸的反應 指導老師：康玉專 小組組員: 楊海晴（15號） 鄭書玉（20號） 鄧嘉怡（21號） 戴詠雅（22號） 2016年5月
目錄 壹、課題研究動機 (3) 貳、課題研究方向 (3) 參、研究試劑和器材 (3) 肆、研究過程（研究步驟、結果與討論） (4) 一、氫離子濃度對反應的影響 (4) 二、去膜的鋁與鹽酸、硫酸反應 (5) 三、C l-對反應的影響……………………………5-6 四、S O42-對反應的影響…………………………6-7 伍、結論…………………………………………………………….7-8
鋁和稀鹽酸、稀硫酸的反應 小組成員：楊海晴(15)、鄭書玉(20)、鄧嘉怡(21)、戴詠雅(22) 壹、課題研究動機 高一的化學課程───鋁及其化合物，是最吸引我們的一環，更令我們感興趣的是鋁的化學實驗。其中我們最好奇的是鋁和稀鹽酸以及和稀硫酸的反應。平常讓人感到難以接觸到的，在化學課堂上較少能夠深究的，我們都很難親身去嘗試到的實驗課程，終於可借此體驗一番，真是令人興奮不已。課餘閒聊之時，突然靈機一觸：為什麼老師再演示實驗過程時會特別強調所用酸的不同濃度。如果做實驗的時候，參與實驗的化學品換作是另一濃度的酸，那麼實驗結果是否會有所變化呢？我們做出來效果是否與老師所演示的差不多呢？於是我們決定以此作為專題報告，開始我們的鋁的實驗大變身──鋁和稀鹽酸，稀硫酸的實驗。 貳、課題研究方向 一、氫離子濃度對反應的影響 二、去膜的鋁與鹽酸、硫酸反應 三、Cl-對反應的影響 四、SO42-對反應的影響 參、研究試劑和器材 一、藥品：鋁片、硫酸、鹽酸、氯化鈉、硫酸鈉 二、器材：試管、燒杯、砂紙、膠頭滴管、鑷子、酒精燈 肆、研究過程（研究步驟、結果與討論） 一、氫離子濃度對反應的影響 (一)、實驗步驟：
硫酸与盐酸的性质对比物理性质：
4 2HCl 化学性质： 1.稀盐酸，稀硫酸具有酸的通性： ①能和指示剂反应。（使紫色石蕊试液变红，使酚酞不变色） ②能和活泼金属反应（排在H前的金属，如铁，锌，镁，铝） 对应训练1：实验室制取氢气，为什么一般选用锌和稀盐酸（或稀硫酸），而不用铁，也不用铝？
③能和碱反应，生成盐和水。（即中和反应）注意：生成盐和水的反应不一定都是中和反 应。例如：2NaOH+CO2 = Na2CO3+H2O ④能和碱性氧化反应。例如：CaO+2HCl = CaCl2+H2O ⑤能和某些盐反应。如：H2SO4+BaCl2 =BaSO4↓+2HCl （注意：该反应是复分解反应，反应物要可溶，生成物要有沉淀，气体或水，这三种物质中的其中一种） 2.浓硫酸的浓度最高可达98%，所以此时浓硫酸具有它独特的性质。 ①强氧化性。 ②脱水性。 如何稀释浓硫酸？ 为什么不能将水倒入浓硫酸中？为什么要用玻璃棒不断搅拌呢？ 1.下列说法不正确的是（） A.稀硫酸能和铁反应 B.稀硫酸具有挥发性 C.稀盐酸不能使酚酞试剂变红色 D.浓盐酸放在空气中浓度会变低 2.下列化学方程式书写正确的是（） A. 2Fe +3 H2SO4 = Fe2(SO4)3 +3H2↑ B. 2HCl + BaO =BaCl2 +H2O C. HCl + AgNO3 =AgCl +HNO3 D．NaOH + CaCl2 =Ca(OH)2+ NaCl 3.我们已经学过，实验室制取二氧化碳用石灰石和稀硫酸反应。实验的反应方程式为： CaCO3+ 2HCl = CaCl2+ H20+ CO2↑请问: (1)能否用浓盐酸代替稀盐酸？ ————————————————————————————
硫酸与盐酸的性质对比 物理性质： 化学性质： 1.稀盐酸，稀硫酸具有酸的通性： ①能和指示剂反映。（使紫色石蕊试液变红，使酚酞不变色） ②能和活泼金属反应（排在H前的金属，如铁，锌） 对应训练1：实验室制取氢气，为什么一般选用锌和稀盐酸（或稀硫酸），而不用铁，也不用铝？ 答案：因为铁和酸反应太慢，铝和酸反应太快，都不易收集。 ③能和碱反应，生成盐和水。（即中和反应）注意：生成盐和水的反 应不一定都是中和反应。例如：2NaOH+CO2 == Na2CO3+H2O ④能和碱性氧化反应。例如：CaO+2HCl == CaCl2+H2O ⑤能和某些盐反应。如：H2SO4+BaCl2 ==BaSO4↓+2HCl （注意：该反应是复分解反应，反应物要可溶，生成物要有沉淀，气体或水，这三种物质中的其中一种） 2.浓硫酸的浓度最高可达98%，所以此时浓硫酸具有它独特的性质。 ①强氧化性。例如：稀硫酸不能和Cu反应，但浓硫酸在加热条件下
可以与铜反应。（初中对这个方程式不作要求，故不写了） ②脱水性。浓硫酸能将有机物(包括木材,布匹,皮肤,糖类等)中的 氢元素和氧元素按照2比1的比例(以水的形式)脱去,最后只剩下碳元素。例如浓硫酸脱去蔗糖中的水，蔗糖变黑色。 ③因为浓硫酸的脱水性和强氧化性，它具有非常的腐蚀性。所以一 旦不慎溅到衣服或皮肤，应马上用布吸干，再用大量水冲洗，最后涂上低浓度的碳酸氢钠溶液。（这是初中的内容，到高中后，老师会讲可以直接用大量水冲洗。具体原因不在此解释。） 对应训练2-1：如何稀释浓硫酸？ 答案：将浓硫酸沿着玻璃棒或烧杯内壁倒入水中，并不断用玻璃棒搅拌。 对应训练2-2：为什么不能将水倒入浓硫酸中？为什么要用玻璃棒不断搅拌呢？ 答案：以为水的密度比浓硫酸小。（98%的浓硫酸的密度大约为1.84g/cm3.如果将水倒入浓硫酸，水无法下沉。但浓硫酸溶于水后发出大量热，会使水飞溅出去。使用玻璃棒是为了防止局部过热，以免液体飞溅。 作业设计： 1.下列说法不正确的是（） A.稀硫酸能和铁反应 B.稀硫酸具有挥发性 C.稀盐酸不能使酚酞试剂变红色 D.浓盐酸放在空气中浓度会变低 2.下列化学方程式书写正确的是（）