Hassett-Tschinkel correspondence: Modality and projective hypersurfaces

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by  Ivan V. Arzhantsev

HASSETT-TSCHINKEL CORRESPONDENCE: MODALITY AND
PROJECTIVE HYPERSURFACES
IVANV.ARZHANTSEV AND ELENAV.SHAROYKO
Abstract. In [11], B. Hassett and Yu. Tschinkel introduced a remarkable correspondence be-
tweengenericallytransitiveactionsofthecommutativeunipotentalgebraicgroupGna andfinite-
dimensional local algebras. In this paper we develop Hassett-Tschinkel correspondence and
calculate modality of generically transitive Gna-actions on projective spaces, classify actions of
modality one, and characterize generically transitive Gna-actions on projective hypersurfaces of
givendegree. Inparticular,actions ondegenerate projectivequadricsarestudied.
0
1
0
2 Introduction

g
The theory of toric varieties plays an important role in modern Geometry, Algebra, Topology
u
and Combinatorics. It is caused by a beautiful description of toric varieties in terms of convex
A
geometry. Actualeffortstogeneralizesuchadescriptiontootherclassesofobjectsareverynatural.

8 LetusrecallthatanirreduciblenormalalgebraicvarietyX iscalledtoricifthereisaregularaction
1 of an algebraic torus T on X with an open orbit, see, for example, [10]. This definition can be

generalized in different ways. One possibility is to consider arbitrary torus actions on algebraic
]
G varieties. Recentlyasemi-combinatorialdescriptionofsuchactionsintermsofso-calledpolyhedral
A divisorslivingonvarietiesofsmallerdimensionwasintroducedin[1]and[2]. Anothervariantisto
restrict the (complex) algebraic torus action on a toric variety to the maximal compact subtorus
.
h (S1)n, to axiomatize this class of (S1)n-actions, and to consider such actions on wider classes
t
a of topological spaces. This is an active research area called Toric Topology, see [7]. Further,
m one may consider algebraic actions with an open orbit replacing the torus T with a non-abelian
[ connected reductive algebraic group G. The study of generically transitive G-actions leads to the
theory of equivariant embeddings of homogeneous spaces G/H, where H is an algebraic subgroup
2
of G. This theory is well-developed. In particular, there are classes of homogeneous spaces with
v
combinatorial description of embeddings. These are the so-called spherical homogeneous spaces
4
7 or, more generally, homogeneous spaces of complexity not exceeding one. The description of
4 embeddings here is much more complicated than in the toric case, see [14], [18].
1 ItisalsonaturaltoreplacethetorusT withthecommutativeunipotentgroupGn =G ×···×G
a a a
2. (ntimes),whereGa istheadditivegroupofthegroundfieldK. Thetheoryofgenericallytransitive
1 Gn-actionsmaybe regardedasan”additiveanalogue”oftoricgeometry. Butinthiscasewecome
a
9
across principal differences. It is easy to see that a generically transitive action of a unipotent
0
group on an affine variety is transitive [15, Section 1.3]. So, in contrast with the toric case, we
:
v can not cover a variety with generically transitive Gn-action by invariant affine charts. Also it is
a
i
X known that any toric variety contains finitely many T-orbits and that an isomorphism between
toricvarietiesasalgebraicvarietiesprovidestheir isomorphisminthe categoryoftoricvarieties[6,
r
a Theorem 4.1]. In additive case these properties do not hold: one may consider two G2-actions on
a
the projective plane P2 given in homogeneous coordinates as
[z :z :z ]→[z :z +a z :z +a z ]
0 1 2 0 1 1 0 2 2 0
and
a2
[z :z :z ]→[z :z +a z :z +a z +( 1 +a )z ].
0 1 2 0 1 1 0 2 1 1 2 0
2
In the first case, there is a line consisting of fixed points, while for the second action there are
three G2-orbits. Finishing with these negative observations, we turn to a positive side.
a
In [11], an algebraic interpretation of generically transitive Gn-actions on Pn is given. Namely,
a
it is shown there that such actions correspond to local finite-dimensional algebras. The aim of
this paper is to study this correspondence in more details. In Section 1 we recall basic facts
on generically transitive Gn-actions. There is an initial correspondence between cyclic rational
a
faithful (m+1)-dimensional Gn-modules V and isomorphism classes of pairs (R,U), where R is
a
a local associative commutative K-algebra with an m-dimensional maximal ideal m and U is an
ThefirstauthorwassupportedbyRFBRgrants09-01-99648-,09-01-90416-Ukr-f-a,andtheDeligne2004Balzan
PrizeinMathematics.
1
2 I.V.ARZHANTSEV AND E.V.SHAROYKO
n-dimensional subspace of m that generates the algebra R. These data determine a generically
transitive Gn-action on the closure of the orbit Gnhvi in the projectivization Pm of the space V
a a
with v ∈ V being a cyclic vector. Conversely, every generically transitive Gn-action on a normal
a
projective variety arises in this way.
Suppose that the subspace U coincides with the maximal ideal m, or equivalently, n = m.
Here we obtain a generically transitive Gn-action on the projective space Pn. Let us recall that
a
modality ofa Gn-actionis the maximalnumber of parametersina continuousfamily ofGn-orbits.
a a
In Section 2 modality is calculated in terms of the algebra R. Also some estimates on modality
are given. The next section is devoted to classification of Gn-actions on Pn of modality one.
a
The classification results in series of finite-dimensional 2-generated local algebras (Theorem 3.1).
It implies that for every n ≥ 5 there are exactly n+1 generically transitive Gn-actions on Pn of
a
modalityone. Onthecontrary,actionsofmodalitytwoadmitmodulispacesofpositivedimension.
Itbringssomeanalogywiththeembeddingtheoryforhomogeneousspacesofreductivegroupswith
small complexity, cf. [18]. Let us also mention that finite-dimensional 2-generated local algebras
occurinthestudyofpairsofcommutingnilpotentmatriceswithafixedcyclicvector. Latelythere
aremanyworksinthisdirection,see[5]andreferencestherein. Thevarietyofpairsofcommuting
nilpotent n×n matrices with a fixed cyclic vector admits a natural action of the general linear
group. It is shown in [13, Theorem 1.9] that if we consider the canonical map from the punctual
HilbertschemeHilbn(A2)tothesymmetricpowerSymn(A2),thequotientbytheactionmentioned
above is the fibre H[n] over the point np, where p is a point on A2. It would be interesting to
interpret Hassett-Tschinkel correspondence in these terms.
Next,genericallytransitiveGn-actionsonprojectivehypersurfacesarisewhenU isahyperplane
a
in the ideal m. Generically transitive Gn-actions on the non-degenerate quadric Q ⊂ Pn+1 are
a n
describedin[16]. Itisshowntherethatforeverynsuchanactionisuniqueuptoisomorphism,and
the corresponding pair (R,U) is indicated. In Section 4 we characterize pairs (R,U) representing
actions on degenerate quadrics. It turns out that here families (of isomorphism classes) of actions
admit moduli.
LetU beahyperplaneinm. Thenthe pair(R,U)determinesagenericallytransitiveGn-action
a
onahypersurfaceX ⊂Pn+1. InthelastsectionweprovethatthedegreeofX equalsthemaximal
exponent d such that md is not contained in U.
The ground field K is assumed to be algebraically closed and of characteristic zero.
1. Hassett-Tschinkel correspondence
In this section we recall some results from [11], see also [16, § 2]. Let ρ : Gn → GL (K)
a m+1
be a faithful rational representation. The differential defines a representation dρ : g → gl (K)
m+1
of the tangent algebra g = Lie(Gn) and the induced representation τ : U(g) → Mat (K) of
a m+1
the universal enveloping algebra U(g). Since the group Gn is commutative, the algebra U(g)
a
is isomorphic to the polynomial algebra K[x ,...,x ], where g is identified with the subspace
1 n
hx ,...,x i. The algebra R := τ(U(g)) is isomorphic to the factor algebra U(g)/Kerτ, where
1 n
Kerτ = {y ∈ U(g) : τ(y) = 0}. As τ(x ),...,τ(x ) are commuting nilpotent operators, the
1 n
algebraR is finite-dimensional and local. Let us denote by X ,...,X the images of the elements
1 n
x ,...,x in R. Then the maximal ideal of R is m := (X ,...,X ). Cleary, the subspace U :=
1 n 1 n
τ(g)=hX ,...,X i generates the algebra R.
1 n
Assume that Km+1 is a cyclic Gn-module with a cyclic vector v, i.e., hρ(Gn)vi = Km+1. The
a a
subspaceτ(U(g))v isg-andGn-invariant;itcontainsthe vectorv andthereforecoincideswiththe
a
space Km+1. Let I = {y ∈ U(g) : τ(y)v = 0}. Since the vector v is cyclic, the ideal I coincides
with Kerτ, and we get identifications
R∼=U(g)/I ∼=τ(U(g))v =Km+1.
Under these identifications the action of an element τ(y) on Km+1 corresponds to the operator of
multiplication by τ(y) on the factor algebra R, and the vector v ∈Km+1 goes to the residue class
ofunit. Since Gn =exp(g), the Gn-actionon Km+1 correspondsto the multiplication by elements
a a
of exp(U) on R.
Conversely, let R be a local (m+1)-dimensional algebra with a maximal ideal m, and U ⊆ m
be a subspace that generates the algebra R. Fix a basis X ,...,X in U. Then R admits a
1 n
presentation K[x ,...,x ]/I, where I is an ideal in the polynomial algebra K[x ,...,x ]. These
1 n 1 n
data define a faithful representation ρ of the group Gn := exp(U) on the space R: the operator
a
ρ((a ,...,a )) acts as multiplication by the element exp(a X +···+a X ). Since U generates
1 n 1 1 n n
R, one checks that the representationis cyclic with unit in R as a cyclic vector.
Summarizing, we get the following result.
MODALITY AND PROJECTIVE HYPERSURFACES 3
Theorem 1.1. [11, Theorem 2.14]. The correspondence described above establishes a bijection
between
(1) equivalence classes of faithful cyclic rational representations ρ:Gn →GL (K);
a m+1
(2) isomorphism classes of pairs (R,U), where R is a local (m+1)-dimensional algebra with
a maximal ideal m and U is an n-dimensional subspace of m that generates the algebra R.
Remark 1.2. Letρ:Gn →GL (K)beafaithfulcyclicrationalrepresentation. Thesetofcyclic
a m+1
vectorsinKm+1isanopenorbitofacommutativealgebraicgroupCwithρ(Gn)⊆C ⊆GL (K),
a m+1
and the complement of this set is a hyperplane. In our notation, the group C is the extension of
the commutative unipotent group exp(m) by scalar matrices.
Afaithful linearrepresentationρ:Gn →GL (K)determines aneffective actionofthe group
a m+1
Gn ontheprojectivizationPm ofthespaceKm+1. Conversely,letGbeaconnectedaffinealgebraic
a
groupwith the trivial Picardgroup,and X be a normal G-variety. By [12, Section 2.4], every line
bundle on X admits a G-linearization. Moreover, if G has no non-trivial characters, then a G-
linearization is unique. This shows that every effective Gn-action on Pm comes from a (unique)
a
faithful rational (m+1)-dimensional Gn-module.
a
An effective Gn-action on Pm is generically transitive if and only if n = m. In this case the
a
corresponding Gn-module is cyclic. It terms of Theorem 1.1 the condition n = m means U = m,
a
and we obtain the following theorem.
Theorem 1.3. [11, Proposition 2.15] There is a one-to-one correspondence between:
(1) equivalence classes of generically transitive Gn-actions on Pn;
a
(2) isomorphism classes of local (n+1)-dimensional algebras.
In this correspondence we realize the space Pn as the projectivization P(R) of the algebra R,
andthe Gn-actionisgivenasthe actionofthe groupofinvertibleelements ofRby multiplication.
a
The complement of the open Gn-orbit on P(R) is the hyperplane P(m).
a
Remark 1.4. Recall that two elements a and b of an algebra R are associated if there exists an
invertible element c ∈ R such that a = bc. The correspondence of Theorem 1.3 determines a
bijection between Gn-orbits on Pn and association classes of nonzero elements in the algebra R.
a
Remark 1.5. The correspondence of Theorem 1.3 together with classification results from [17]
yields that the number of equivalence classes of generically transitive Gn-actions on Pn is finite if
a
and only if n≤5, see [11, Section 3].
2. Modality of actions on projective spaces
SupposethatanaffinealgebraicgroupGactsregularlyonanirreduciblealgebraicvarietyX. It
iswellknown[15,Section1.4]thatthereisanon-emptyopensubsetW ⊆X consistingofG-orbits
ofmaximaldimension. Byd(G,X)denotethe codimensionofthe orbitGxinX,wherex∈W. It
follows from Rosenlicht’s Theorem [15, Section 2.3] that d(G,X) equals the transcendency degree
of the field K(X)G of rationalinvariantson the variety X. The condition d(G,X)=0 means that
the G-action is generically transitive.
Themodalitymod(G,X)ofaG-actiononX isthemaximalvalueofd(G,Y)overallirreducible
G-invariant subvarieties Y ⊆ X. In particular, we have mod(G,X) = 0 if and only if X contains
a finite number of G-orbits.
Consider an action of the group Gn on the space Pm corresponding to the pair (R,U). Note
a
that the isotropy subalgebra g of an element x∈R coincides with the subspace Ann (x)={y ∈
x U
U : xy =0}. Inparticular,wehaveg =0foreveryx∈R\m. DenotebyC theZariskiclosure
x k,U
of the subset {x ∈ m : dimAnn (x) = k} ⊆ m. Then the typical Gn-orbit on each irreducible
U a
component of C is of dimension n−k. This gives a first interpretation of modality.
k,U
Proposition 2.1. The following equality holds
mod(Gn,Pm)=max(dimC +k)−(n+1).
a k,U
k
InthesequelweconsidergenericallytransitiveGn-actionsonPn,i.e.,assumen=mandU =m.
a
Let us introduce some notation: Ann(x) := Annm(x), Ck := Ck,m and mod(R) := mod(Gna,Pn).
Clearly, 0 ≤ mod(R) ≤ n−1. For each n, Hassett and Tschinkel have proved that the extreme
value 0 is achieved for a unique algebra R, see also Corollary 2.6 below. In Proposition 2.7 we
prove that the extreme value n−1 is also achieved in a unique way.
Example 2.2. Let R=K[x ,...,x ]/I , where the ideal I ⊂K[x ,...,x ] is generatedby
1 s N+1 N+1 1 s
all monomials of degree N+1. The annihilator of a polynomial f(X ,...,X )∈R coincides with
1 s
4 I.V.ARZHANTSEV AND E.V.SHAROYKO
the linear span of all monomials of degree ≥N +1−k, where k is degree of the lowestterm of f.
Therefore all non-empty subsets C are linear spans of monomials of degree greater or equal to a
k
given value, and
N s+i−1 [N2] s+i−1
mod(R)= − −1.
(cid:18) s−1 (cid:19) (cid:18) s−1 (cid:19)
i=X[N+1] Xi=1
2
In particular, N =1 and s=n implies mod(R)=n−1.
The operator of multiplication by x ∈ mi maps mj in mi+j. Thus dimAnn(x) ≥ dimmj −
dimmi+j, and we get
Lemma 2.3. mod(R)≥max (dimmi+dimmj −dimmi+j)−(n+1).
i,j
Set r := dimmi −dimmi+1. In particular, r = 1. Suppose mN 6= 0 and mN+1 = 0. Then
i 0
n=r +r +···+r . Using Lemma 2.3 with j =1, we obtain
1 2 N
Proposition 2.4. mod(R)≥max r −1.
i i
Example 2.2 shows that the modality can significantly exceed the value max r −1.
i i
Corollary2.5. Ifmod(R)=l,thenthealgebraRcanbegeneratedbyl+1elementsanddimmN ≤
l+1.
Corollary 2.6. [11,Proposition3.7] For any n>0 there exists a unique (n+1)-dimensional local
algebra of modality zero; it is isomorphic to K[x]/(xn+1).
Proof. By Corollary 2.5, an algebra of modality zero is generated by one element. (cid:3)
Letusrecallthatthesequence(r ,r ,...,r )iscalledtheHilbert-Samuelsequenceofthealgebra
0 1 N
R. This sequence does not determine modality of the algebra. Indeed, consider the algebras
K[x,y]/(x4,xy,y3) and K[x,y]/(x4,y2,x2y).
They share the same Hilbert-Samuel sequence (1,2,2,1). In the next section it is shown that the
first algebra has modality one, while the second one has modality two.
The inequality mod(R) ≥ r −1 can be strengthened as mod(R) ≥ dimAnn(m)−1, where
N
Ann(m) = {x ∈ m : yx = 0 for all y ∈ m}. The subset P(Ann(m)) coincides with the set of
Gn-fixed points in Pn. It is known that a local finite-dimensional algebra R is Gorenstein if and
a
only if dimAnn(m) = 1 [8, Proposition 21.5]. Such algebras are assigned to generically transitive
Gn-actions on Pn with a unique fixed point.
a
Proposition2.7. Foranyn>0thereexistsaunique(n+1)-dimensionallocalalgebra ofmodality
n−1; it is isomorphic to K[x ,...,x ]/(x x , 1≤i≤j ≤n).
1 n i j
Proof. Suppose there is a Gn-action on Pn with an (n−1)-dimensional continuous orbit family.
a
Then any orbit from this family is just a Gn-fixed point. The complement to the open orbit in Pn
a
is the hyperplane P(m), so all the points in this hyperplane should be Gn-fixed. This condition is
a
equivalent to the equality m2 =0. (cid:3)
Remark 2.8. Remark1.4 allowsus to describe mod(R) as the maximalnumber ofparametersina
continuous family of principal ideals of the algebra R. Such families can be represented as locally
closed subsets in the Grassmannians of subspaces of R of appropriate dimension.
Proposition 2.9. If I is a proper ideal of the algebra R, then mod (R/I)≤ mod (R).
Proof. The canonical homomorphism R→R/I induces a homomorphism of groups
Gn :=exp(m)→Gr :=exp(m/I), where r =dim(R/I)−1,
a a
and a surjective morphism of varieties P(R)\P(I)→P(R/I). The latter morphism is equivariant
with respect to the Gn- and Gr-actions on P(R) and P(R/I) respectively. Assume that we have a
a a
continuousk-parameterfamilyofGr-orbitsonP(R/I). Someirreduciblecomponentofthe inverse
a
image of this family projects to the family dominantly. Then typical Gn-orbits on this component
a
form at least k-parameter family on P(R)\P(I). This completes the proof. (cid:3)
Wefinishthissectionbyintroducingacombinatorialwaytoestimatemodalityofalocal(n+1)-
dimensional algebra K[x ,...,x ]/I, where I is a monomial ideal. Denote by N the set of non-
1 s
constant monomials in x ,...,x that do not belong to I. To each subset L⊆N assign a subset
1 s
S(L) = {a ∈ N : ab ∈ I for any b ∈ L}. The subset S(L) consists of monomials annihilating all
the elements of the factor algebra with support at L. It implies the inequality
mod(R)≥max(|L|+|S(L)|)−(n+1),
L
which can be strict. For example, with the algebra K[x ,x ]/(x2,x2) we get 1>0.
1 2 1 2
MODALITY AND PROJECTIVE HYPERSURFACES 5
3. Algebras of modality one
In this section we classify local algebras corresponding to generically transitive Gn-actions on
a
Pn of modality one.
Theorem 3.1. Local finite-dimensional algebras of modality one form the following list:
A =K[x,y]/(xa+1,yb+1,xy), a≥b≥1; B =K[x,y]/(xy,xa−yb), a≥b≥2;
a,b a,b
C =K[x,y]/(xa+1,y2−x3), a≥3; C1 =K[x,y]/(xa+1,y2−x3,xay), a≥3;
a a
C2 =K[x,y]/(xa+1,y2−x3,xa−1y), a≥3; C3 =K[x,y]/(y2−x3,xa−2y), a≥4;
a a
D =K[x,y]/(x3,y2); E =K[x,y]/(x3,y2,x2y).
These algebras are pairwise non-isomorphic.
Proof. It follows from Corollary 2.6 that all algebras of Theorem 3.1 have positive modality. Let
us show that their modality does not exceed one.
Every principal ideal of the algebra A is generated by some element
a,b
α xk+α xk+1+...+β ys+β ys+1+... = (α xk+β ys)(1+α]x+...+β]y+...).
k k+1 s s+1 k s k+1 s+1
It is easy to reduce the proof to the case α 6= 0, β 6= 0. The second factor is invertible, so any
k s
familyofprincipalidealsinA isatmostone-parameter. ByRemark2.8,modalityofthealgebra
a,b
A does not exceed one. The algebra B is a homomorphic image of A , and Proposition 2.9
a,b a,b a,b
implies that modality of B does not exceed one.
a,b
For other types of algebras the following lemma is useful.
Lemma3.2. Supposethatabasisv ,...,v oftheidealmofalocalalgebraRsatisfiesthefollowing
1 n
conditions:
(i) the element v does not belong to the principal ideal (v ) for every s≥1;
s s+1
(ii) for all v and v , p ≥ 2, there exists a vector w ∈ m such that v = v w and
s s+p s,p s+p s s,p
v w =0 or v , r ≥s+p+1.
s+1 s,p r
Then the modality of the algebra R does not exceed one.
Proof. Consider a continuous family {(b ),γ ∈ Γ} of principal ideals in the algebra R. Each
γ
element b is defined up to multiplication by an invertible element of R, and we can assume that
γ
there exists s≥1 such that any element b can be written as
γ
b = v + α v .
γ s p,γ s+p
Xp≥1
Let us show that there exists an element
c =1+β w +...+β w
γ 2,γ s,2 n−s,γ s,n−s
such that b =(v +α v )c . Indeed, write down the equality
γ s 1,γ s+1 γ
v +α v +...+α v = (v +α v )(1+β w +...+β w )
s 1,γ s+1 n−s,γ n s 1,γ s+1 2,γ s,2 n−s,γ s,n−s
and consider the coefficients at v ,...,v on the right. They are equal to α ,...,α
s+2 n 2,γ n−s,γ
respectively, so it is easy to find the values of β ,...,β successively. The element c is
2,γ n−s,γ γ
invertible, hence our family of ideals {(b ),γ ∈ Γ} is at most one-parameter and the statement
γ
follows from Remark 2.8. (cid:3)
Let us indicate a basis satisfying the conditions of Lemma 3.2 for the algebra C :
a
x, y, x2, xy, x3, x2y, ...,xa, xa−1y, xay,
and for the algebra D:
x, y, x2, xy, x2y.
The algebras of types C1, C2, C3 are the factor algebras of C and E is a factor algebra of D, so
a a a a
their modality cannot exceed one.
Now we prove that any algebra of modality one is included in the list of Theorem 3.1.
Lemma 3.3. Modality of the algebra H :=K[x,y]/(x4,y2,x2y) is greater than one.
Proof. One checks that typical G5-orbits on the projective 3-space P(hx2,x3,y,xyi) are one-
a
dimensional. (cid:3)
6 I.V.ARZHANTSEV AND E.V.SHAROYKO
Lemma 3.4. Let R=K[x,y]/I, where the ideal I is generated by polynomials not containing any
of the terms x,x2,x3,y,xy. Then modality of R is greater than one.
Proof. By assumptions, the algebra H is a factor algebra of R. (cid:3)
LetRbealocalfinite-dimensionalalgebraofmodalityone. UsingCorollary2.5,onecanassume
that it has the form K[x,y]/I, where every term of a polynomial from I has degree ≥2. Consider
the algebra R′ =R/m3. It is easy to see that there exists a linear change of variablestaking R′ to
one of the following non-isomorphic algebras:
K[x,y]/(x2,xy,y2), K[x,y]/(x3,xy,y2), K[x,y]/(x2,y2), K[x,y]/(x3,xy,y3), K[x,y]/(x3,x2y,y2).
Case 1: R′ =K[x,y]/(x2,xy,y2). In this case we have m2 =m3, so R=R′ =A .
1,1
Case 2: R′ = K[x,y]/(x3,xy,y2). In this case xy,y2 ∈ m3 and the factor space mi/mi+1
is spanned by the class of the element xi with i ≥ 3. The elements of the form xi are linearly
independentinR,hencetheelements1,y,x,x2,...,xa,a≥2,formabasisofthisalgebra. Assume
that xy = α xi. Replace the basis element y with y′ = y− α xi−1 to obtain xy′ = 0
i≥3 i i≥3 i
and (y′)2 =P β xj. Suppose that the last expression is not zPero and β is the first nonzero
j≥3 j s
coefficient. IPf s < a, then multiplying the equality by xa−s, we get xa = 0, a contradiction. It
means that (y′)2 = 0 or (y′)2 = xa, and the algebras A (a ≥ 2) and B (a ≥ 3) appear.
a,1 a,2
Algebras of the second type are Gorenstein unlike the others. Algebras of the same type with
different indices have different dimensions.
Case 3: R′ =K[x,y]/(x2,y2). In this case x2,y2 ∈m3 and everymonomialof degree three is in
m4, hence m3 =m4 =0, R=R′. It is easy to check that the algebra R′ is isomorphic to B .
2,2
Case 4: R′ = K[x,y]/(x3,xy,y3). In this case we have xy ∈ m3 and R is a linear span of the
elements 1,x,...,xa,y,...,yb, a ≥ b ≥ 2. Assume that xy = α xi + β yj. Let us
i≥3 i j≥3 j
show that there exists a change of variables providing the conditPion xy = 0. IPndeed, take a basis
x′ =x+ φ yk and y′ =y+ ψ xs and write down
k≥2 k s≥2 s
P P
x′y′ = α xi+ β yj + φ yk+1+ ψ xs+1+ φ ψ xsyk.
i j k s k s
Xi≥3 Xj≥3 kX≥2 Xs≥2 kX,s≥2
Here the monomial xsyk is decomposed as the sum of terms xi and yj of degree at least k+s+2
each. Now one can consider the coefficients at x3, y3, x4, y4,... in the expression for x′y′ and
obtain ψ , φ , ψ , φ ,... one by one. Therefore, we can further assume that xy =0.
2 2 3 3
Supposethereisanelementξ xr+···+ξ xa+µ yp+···+µ yb, ξ 6=0,intheidealI. Ifr<a,
r a p b r
thenwecanmultiplybothpartsbyxa−r andcometoacontradictionwiththeconditionxa 6=0.In
the same way it can be shown that p=b. Thus the algebras A , a≥b≥2, and B , a≥b≥3,
a,b a,b
appear. The inequalities are caused by linear independency of the classes of the elements x2 and
y2 in m2/m3. The algebrasfrom the second family are Gorenstein unlike the others. The algebras
of the same type with different indices have different Hilbert-Samuel functions.
Case 5: R′ = K[x,y]/(x3,x2y,y2). In this case we have y2 = α xi + β xjy. A
i≥3 i j≥2 j
change y′ =y+ p≥2φpxp leads to the equation P P
P
(y′)2 = α xi+ β xjy+2 φ xpy+ φ φ xp+s.
i j p p s
Xi≥3 Xj≥2 Xp≥2 pX,s≥2
Settingthecoefficientsatx2y,x3y,... equalzeropermitstoobtainφ ,φ ,... onebyone. Therefore
2 3
one can assume that y2 = α xi. It is easy to see that there exists a root of any integer order
i≥3 i
from an element α +α xP+α x2 +..., where α 6= 0, in R. If y2 = xe(α +α x+...) and
0 1 2 0 e e+1
α 6= 0, then replace y by v−1y, where v2 = α +α x+..., to obtain y2 = xe. Now there are
e e e+1
the variants y2 = 0 or y2 = xe. If the elements xa and xcy are nonzero and xa+1 = xc+1y = 0,
then a ≥ c ≥ 1 a ≥ 2 (since the elements x2 and xy are not in m3). Also the second possibility
requires the restrictions a≥e≥3 and c+e≥a, since y2xc+1 =xc+e+1 =0.
Case 5.1: y2 = 0. There are also the conditions xa+1 = 0 and xc+1y = 0. If there is no
other relation in R, then by Lemma 3.4 we obtain the algebras D and E. Suppose that we have
another equality ξ xi + µ xjy = 0. Then i ≥ 3 and j ≥ 2, since the elements x
i≥i0 i j≥j0 j 0 0
andy arelinearlyPindependentPmod m2, andthe elements x2 andxy arelinearly independent mod
m3. If all ξi = 0 (resp. µj = 0), then we obtain a contradiction multiplying by xc−j0 (resp. by
xa−i0). If i0 ≤ c, then we get a contradiction multiplying by y. Hence i0 ≥ c+1. Moreover, if
a−i 6= c−j , then multiplying by an appropriate power of x again leads to a contradiction. It
0 0
meansthati =a−dandj =c−dforsomed≥0;herea−d>c. Assumethatdismaximalover
0 0
all relations. So there is a relation xa−dχ −xc−dχ y = 0, where χ = α +α x+..., α 6= 0,
1 2 i i0 i1 i0
MODALITY AND PROJECTIVE HYPERSURFACES 7
or equivalently, xa−d−xc−dχy = 0, where χ := χ−1χ . The replacement of y by χy provides the
1 2
relationxa−d−xc−dy =0. Onecansubtractthe relationofthis formfromanyotherrelationwith
correspondinglowertermstocancelthelowertermsortoobtainacontradiction. FromLemma3.4
and the inequality a−d > c > d+1 ≥ 1 it follows that a−d = 3 and c = 2. In this case d = 0,
a=3, c=2, and our algebra can be written as K[x,y]/(y2,x3y,x3−x2y). Replace the variable x
by −3x+y to show that this algebra is isomorphic to D.
Case 5.2: here R is a factor algebraof the algebraK[x,y]/(xa+1,y2−xe,xc+1y). If R coincides
withthisalgebra,thenLemma3.4reducestheprooftothecasee=3. Theinequalitiesa≥c≥a−e
imply four possible variants: c=a, a−1, a−2, a−3. In the last case it follows from c≥1 that
a≥4 and we get the algebras C , C1, C2 and C3 respectively.
a a a a
Assume there is another relation ξ xi + µ xjy = 0. As in the previous case one
i≥i0 i j≥j0 j
shows that i0 ≥ 3, j0 ≥ 2 and a−iP0 = c−j0 =Pd. Again set the value of d maximal over all
relations; thus the equation of described form implies all the relations in the algebra R. Multiply
theequationbyy toobtainanotherequationwithlowertermsxc−d+e andxa−dy. Itiscompatible
with the others whenever there are extra restrictions: two inequalities c−d+e>a and a−d>c
or one equality a−(c−d+e)=c−(a−d). The last formula is equivalent to e=2(a−c). Here
e ≥ 4, hence a−c ≥ 2. It means that a−d ≥ c−d+2; therefore we have a−d ≥ 4, c−d ≥ 2.
From Lemma 3.4 it follows that there are no algebras of modality one in this case.
Thus the restrictions c−d+e > a and a−d > c hold. Lemma 3.4 reduces the remainder to
the following cases:
Case 5.2.1: e=3. From the inequalities c+3−d>a and a−d>c it follows that 3−2d>0,
hence d = 0,1. If d = 1, then c+2 > a > c+1, and we come to a contradiction. If d = 0, then
c+3>a>c; in other words,we have c=a−1 or c=a−2. If the second possibility holds, then
a=c+2≥2+d+2=4. Denote the obtainedalgebrasby D1 and D2 respectively. We are going
a a
to prove that they are isomorphic to C and C3.
a−1 a
Case 5.2.1.1: Da1 ∼= Ca−1 (a ≥ 4) and D31 ∼= D. Consider the following variable change in
C : x′ = ux+vy, y′ = wx2 +zy, where u,v,w,z are some invertible polynomials in x. The
a−1
first required equality (x′)ay′ =0 holds automatically. The second condition (x′)a =(x′)a−1y 6=0
gives an equality for the constant terms: av = z . The third relation (x′)3 = (y′)2 provides the
0 0
equation
x3(u3−z2)+x4(3uv2−w2)+x2y(3u2v−2wz)+x3yv3 =0.
Since the u,v,w,z do not contain y, one obtains all the coefficients of u,v,w,z considering the
coefficients at x3, x2y, x4, x3y,.... Thus we determine the required generators and prove the
isomorphism.
Case 5.2.1.2: D2 ∼= C3. The proof is similar. Consider x′ = ux+ vy and y′ = wx2 + zy
a a
in C3, where u,v,w,z are some invertible polynomials in x. The condition (x′)a−1y′ = 0 holds
a
automatically; the relation (x′)a = (x′)a−2y 6= 0 results in the equality for the constant terms
u3 =u w +(a−2)v z ; the last equality (x′)3 =(y′)2 provides the relation
0 0 0 0 0
x3(u3−z2)+x4(3uv2−w2)+x2y(3u2v−2wz)+x3yv3 =0,
which determines u,v,w,z like in the previous case.
Case 5.2.2: a−d=3. Here we obtain a=3, hence e=3, and this case reduces to 5.2.1.
Now we have to prove that the algebrasC , C1, C2, C3, D and E are not pairwise isomorphic.
a a a a
ThealgebrasC , C3 andD areGorenstein. To provethattheyarenotisomorphicletus calculate
a a
their dimensions:
dimC = 2(a+1) ≥ 8, dimC3 = 2a−1 ≥ 7, dimD = 6.
a a
For non-Gorenstein algebras dimensions are
dimC1 = 2a+1 ≥ 7, dimC2 = 2a ≥ 6, dimE = 5.
a a
This completes the proof of Theorem 3.1. (cid:3)
Corollary 3.5. For n ≥ 5 there are (n+1) pairwise non-isomorphic local (n+1)-dimensional
algebras of modality one, while for n = 2 we have 1 algebra, for n = 3 there are 2 algebras, and
for n = 4 there are 4 algebras. In particular, for any n > 0 the number of equivalence classes of
generically transitive Gn-actions of modality one on Pn is finite.
a
Letus show thatthere isaninfinite family ofpairwisenon-isomorphic7-dimensionallocalalge-
brasofmodalitytwo. ConsideralgebraswithHilbert-Samuelsequence(1,3,3). Heremultiplication
isdeterminedbyabilinearsymmetricmapm/m2×m/m2 →m2. Suchmapsforma18-dimensional
space;thegroupGL(3)×GL(3)actsherewithaone-dimensionalinefficiencykernel. Itmeansthat
8 I.V.ARZHANTSEV AND E.V.SHAROYKO
there are infinitely many generic pairwise non-isomorphic algebras of this type. Let us prove that
modalityofagenericalgebraequalstwo. ByProposition2.1,mod(R)=max (dimC +k)−(n+1);
k k
heretheannihilatorofagenericpointx∈mcoincideswithm2,sok =3andC =m. Whenk =4
3
(resp. 5), the dimension of C decreases by 1 (resp. at leastby 2). Finally, for k =6 the space C
k k
coincides with m2. One concludes that mod(R)=(6+3)−7=2.
Note that for every monomial ideal I the finite-dimensional algebra K[x,y]/I determines the
YoungdiagramwithboxesassignedtoallmonomialsnotinI. Conversely,toeveryYoungdiagram
one assigns a local finite-dimensional monomial algebra with two generators. The diagrams (n)
and (1,...,1) correspond to modality zero.
q
q
q qq q
By Theorem 3.1, diagrams of modality one are exactly (s,1,...,1), (2,2), (3,2), (2,2,1), (3,3)
and (2,2,2).
q qq
q
q
q
ItwouldbeinterestingtocalculatethemodalityforanarbitraryYoungdiagram. Anotherintrigu-
ing problem is to connect Theorem 3.1 with the classification of singularities of modality one in
the sense of [3].
4. Gn-actions on degenerate quadrics
a
We return to the settings of Section 1. Given the projectivization Pm of a rational Gn-module
a
andapointx∈Pm,theclosureoftheorbitGn·xisaprojectivevarietywithagenericallytransitive
a
Gn-action. Conversely, any normal projective Gn-variety can be equivariantly embedded into the
a a
projectivization of a Gn-module [15, Section 1.2]. If the action is generically transitive, then one
a
may assume that this is the orbit closure of a cyclic vector. Therefore any generically transitive
Gn-action on a normal projective variety is determined by a pair (R,U), see Theorem 1.1. This
a
time the correspondence is not bijective, since the orbit closure is not necessarily normal and
the equivariant embedding of a projective variety into the projectivization of a Gn-module is not
a
unique. For example, the pair (K[x]/(xm+1),hxi) determines a standard G -action on P1 for all
a
m≥1.
LetRbealocal(m+1)-dimensionalalgebrawithamaximalidealmandU beann-dimensional
subspaceof m thatgeneratesthe algebraR. A pair (R,U)is calledanH-pair if U is a hyperplane
in m, or, equivalently, m = n+1. Such pairs correspond to generically transitive Gn-actions on
a
hypersurfaces in Pn+1. The degree of the hypersurface is called the degree of the H-pair. An
H-pair of degree 2 is called quadratic.
Example 4.1. Consider the H-pair (R,U) = (K[x]/(x5),hx,x2,x4i). Denote by X , X and X
1 2 3
the images of the elements x, x2 and x4 in the algebra R. We have
a2 a2 a3 a2a a4
exp(a X +a X +a X )=1+a X +a X +a X + 1X + 2X +a a X3+ 1X3+ 1 2X + 1X ,
1 1 2 2 3 3 1 1 2 2 3 3 2 2 2 3 1 2 1 6 1 2 3 24 3
and the G3-orbit of the line h1i has the form
a
a2 a3 a2 a2a a4
1 : a : a + 1 : a a + 1 : a + 2 + 1 2 + 1 ; a ,a ,a ∈K .
1 2 1 2 3 1 2 3
(cid:26)(cid:20) 2 6 2 2 24(cid:21) (cid:27)
In homogeneous coordinates [z : z : z : z : z ] on P4 our pair determines a hypersurface given
0 1 2 3 4
by the equation:
3z2z −3z z z +z3 =0.
0 3 0 1 2 1
Consider the quadric Q(n,k)⊂Pn+1, k ≥1, defined by equation q =0, where q is a quadratic
formofrankk+2. Inthesenotationthenon-degeneratequadricQ ⊂Pn+1 isQ(n,n). Itisproved
n
MODALITY AND PROJECTIVE HYPERSURFACES 9
in [16] that the quadric Q admits a unique (up to equivalence) generically transitive Gn-action
n a
corresponding to the pair (A ,U ), where
n n
A =K[y ,...,y ]/(y y , y2−y2; i6=j) and U =hY ,...,Y i
n 1 n i j i j n 1 n
as n > 1, and A = K[y ]/(y3), U = hY i. Here by Y we denote the image of the element
1 1 1 1 1 i
y in A . We describe below the H-pairs that determine generically transitive Gn-actions on
i n a
degeneratequadrics. An algebrahomomorphismφ:R→R′ is calleda homomorphism of H-pairs
φ : (R,U) → (R′,U′) if φ(U) = U′. If φ is a homomorphism of H-pairs, then it is a surjective
homomorphism of algebras.
Proposition 4.2. An H-pair (R,U) determines a generically transitive Gn-action on the quadric
a
Q(n,k) if and only if there exists a homomorphism of H-pairs φ:(R,U)→(A ,U ).
k k
Proof. Suppose that a pair (R,U) determines a generically transitive Gn-action on Q(n,k). Let q
a
be the corresponding quadratic Gn-invariant form and B the associated bilinear symmetric form.
a
The operator of multiplication by an element a ∈ U is skew-symmetric with respect to the form
B, i.e., B(ab,c)+B(b,ac)=0 for every b,c∈R. If an element c is in the kernel I of the form B,
then the first term is zero, and ac ∈ I. The subspace U generates the algebra R, so I is an ideal
in R.
Let us show that I ⊆ U. Otherwise one has m = U +I. Putting b = c = 1 in the condition
of skew-symmetry, we get B(1,U) = 0 and hence B(1,m) = 0. For every u ,u ∈ U one has
1 2
B(u ,u )+B(1,u u )=0. It means that B restricted to U is zero. Note that B(1,1)= 0, since
1 2 1 2
the unit corresponds to a point on the quadric. But the form B should be of rank at least 3, a
contradiction.
Theformq inducesanon-degeneratequadraticformq onthefactoralgebraR/I. Thecondition
ofskew-symmetryshowsthatqisGk-invariant,wheretheGk-actionisgivenbythepair(R/I,U/I).
a a
By [16, Theorem 3] this pair is isomorphic to (A ,U ), and the homomorphism we need is the
k k
projection (R,U)→(R/I,U/I).
Conversely,assume that φ:(R,U)→(A ,U ) is a homomorphismof H-pairs. One can lift the
k k
non-degenerate Gk-invariant form q on A to a Gn-invariant form q on R by putting the kernel I
a k a
of the form q equals Ker(φ). We know that the H-pair (R,U) determines a generically transitive
Gn-action on some hypersurface and that q(1)=0; so this hypersurface is the quadric q =0. (cid:3)
a
Corollary 4.3. If φ : (R,U) → (R′,U′) is a homomorphism of H-pairs and the H-pair (R′,U′)
is quadratic, then the H-pair (R,U) is quadratic as well.
Theorem 4.4. An H-pair (R,U) is quadratic if and only if m3 ⊂U.
Proof. If m3 ⊂U, then (R/m3,U/m3) is an H-pair. If we prove that any pair (R,U) with m3 =0
is quadratic,thenTheorem4.4followsfromCorollary4.3. The algebraR canbe decomposedinto
the direct sum of its subspaces: R = h1i⊕U ⊕hµi, where µ ∈ m2. Define a bilinear symmetric
form B on the space R by B(1,1)= 0, B(1,µ) = −1, B(1,U)= 0, and for every x,y ∈ m obtain
the value B(x,y) from the equality xy = u+B(x,y)µ, where u ∈ U. In this case the rank of
the form B is at least 3. Let us check that the operator of multiplication by an element u∈ U is
skew-symmetricwith respect to B. Fix a basis e =1, e ,...,e ∈U and e =µ in the algebra
0 1 n n+1
R. It suffices to show that B(ue ,e )+B(e ,ue ) = 0 for all i ≤ j. Here both terms are zeroes
i j i j
apart from the case i=0, j =1,...,n, and B(ue ,e ) is the coefficient at µ in the decomposition
0 j
of the element ue , while B(e ,ue ) is opposite to the coefficient at µ for the same element.
j 0 j
Conversely, suppose that an H-pair (R,U) is quadratic. Then for some k there exists a homo-
morphism of H-pairs φ : (R,U) → (A ,U ). If m3 is not contained in U, then its image φ(m3)
k k
is not contained in U , hence φ(m3) 6= 0. But in A the cube of the maximal ideal is zero, a
k k
contradiction. (cid:3)
Corollary 4.5. Consider a homomorphism φ:(R,U)→(R′,U′) of H-pairs. The H-pair (R,U)
is quadratic if and only if the H-pair (R′,U′) is quadratic.
Let us take a closer look to pairs (R,U) determining generically transitive Gn-actions on the
a
quadric Q(n,n−1). There is a homomorphism φ : (R,U) → (A ,U ) described in Propo-
n−1 n−1
sition 4.2. Its kernel I should be one-dimensional, we have mI = 0 and Ann(m) = hI,Ci,
where C ∈ m is an element with φ(C) = Y2. The multiplication determines a bilinear map
1
(U/I)×(U/I) → Ann(m). Let us fix a basis vector P in I and obtain two bilinear symmetric
forms B and B on U/I:
1 2
xy =B (x+I,y+I)P +B (x+I,y+I)C for all x,y ∈U.
1 2
10 I.V.ARZHANTSEV AND E.V.SHAROYKO
The form B is non-degenerate. Not the vectors P and C, but the line hB i and the linear span
2 2
hB ,B i are defined uniquely. If B and B are proportional, one can obtain B =0 by changing
1 2 1 2 1
the basis; in this case we get the pair
R=K[x ,...,x ,p]/(x x ,x2−x2,x p,p2;i6=j), U =hX ,...,X ,Pi.
1 n−1 i j i j i 1 n−1
Otherwise the flag hB i ⊂ hB ,B i in the space of bilinear forms on U/I is well-defined. In an
2 1 2
appropriate basis the form B is represented by the identity matrix, and the matrix of the form
2
B is diagonal. One can assume that the secondmatrix has 1 and0 as first two diagonalelements
1
and other n−3 diagonal elements determine the parameters for isomorphism classes of the pairs.
Hencethereisaninfinitefamilyofpairwisenon-equivalentgenericallytransitiveG4-actionsonthe
a
quadric Q(4,3). Set the quadric Q(4,3) by the equation z2+z2+z2 =2z z in P5; the action of
1 2 3 0 5
the element (a ,a ,a ,a ) at the point [z :z :z :z :z :z ] is given by the formula
1 2 3 4 0 2 2 3 4 5
a2+a2t+2a
[z :a z +z :a z +z :a z +z : 2 3 4z +a z +ta z +z :
0 1 0 1 2 0 2 3 0 3 0 2 2 3 3 4
2
a2+a2+a2
: 1 2 3z +a z +a z +a z +z ]
0 1 1 2 2 3 3 5
2
dependingontheparametert∈K. IftheformsB andB areassignedtothematricesdiag(1,0,t)
1 2
and E, where t6=0,1, then the flag hB i⊂hB ,B i determines the value t up to transformations
2 1 2
t→ 1 andt→1−t. Itmeansthatdifferentvaluest andt oftheparameterdetermineequivalent
t 1 2
actions if and only if
(t2−t +1)3 (t2−t +1)3
1 1 = 2 2 .
t2(1−t )2 t2(1−t )2
1 1 2 2
The values t=0 and t=1 determine one more class of equivalent actions.
5. Degree of a hypersurface
The following result generalizes Theorem 4.4.
Theorem 5.1. Let X be the closure of a generic orbit of an effective Gn-action on Pn+1 and
a
(R,U) be the corresponding H-pair. Then the degree of the hypersurface X equals the maximal
exponent d such that the subspace U does not contain the ideal md.
Proof. LetN bethemaximalexponentwithmN 6=0anddbethemaximalexponentwithmd 6⊂U.
Consider the flag U ⊆ U ⊆ ··· ⊆ U = U, where U = mi∩U, and construct a basis {e } in
N N−1 1 i i
U coordinatedwiththis flag. Addavectore∈md\U toobtaina basisinm. Foranonzerovector
v ∈ m the maximal number l such that v ∈ ml is called the weight of the vector and is denoted
by ω(v). For example, ω(e) = d. One may assume that the weights of the vectors e ,...,e do
1 n
not decrease, that the vectors e ,...,e are the basis vectors of weight 1, and that the vectors
1 s
e ,...,e are the basis vectors of weight < d. The hypersurface X coincides with the closure of
1 k
the projectivization of the set
{exp(a e +···+a e ) : a ,...,a ∈K}.
1 1 n n 1 n
Supposethatexp(a e +···+a e )=1+z e +···+z e +ze. Thenz =a +f (a ,...,a ),where
1 1 n n 1 1 n n i i i 1 ri
the weights of e ,...,e do not exceed ω(e ). In particular, z = a ,...,z = a . Assume that
1 ri i 1 1 s s
i>s. For any term αaj1...ajri, α∈K×, of the polynomial f one has j ω(e )+···+j ω(e )≤
1 ri i 1 1 ri ri
ω(e ).
i
Observe that a = z −f (a ,...,a ). One can show by induction that any element a , p =
i i i 1 ri p
1,...,r , can be expressed as a polynomial F of degree ≤ ω(e ) in z ,...,z ,z . It means
i p p 1 rp p
that a can be expressed as a polynomial F of degree ≤ ω(e ) in z ,...,z ,z . Finally, we have
i i i 1 ri i
z =f(a ,...,a ), where for every term αaj1...ajk, α∈K×, one has j ω(e )+···+j ω(e )≤d.
1 k 1 k 1 1 k k
Replace each a , i = 1,...,k by its expression in z ,...,z ,z and obtain z = F(z ,...,z ).
i 1 ri i 1 k
Therefore the degree of every term of the polynomial F does not exceed d.
For any polynomial G(y ,...,y ) of degree r one can multiply each term of degree q by zr−q
1 m 0
to get a homogeneous polynomial that we denote by HG(z ,y ,...,y ). The closure of the orbit
0 1 m
Gnh1i is given by the equation HF(z ,z ,...,z ,z), where F(z ,...,z ,z) = z −F(z ,...,z ).
a 0 1 k 1 k 1 k
Here z denotes the first coordinate in the basis {1,e ,...,e ,e} of the algebra R.
0 e 1 ne
We have to show that there is a term of degree d in the polynomial F. We use induction on
n. If n = 1, then the corresponding pair is (R,U) = (K[y]/(y3),hyi). Here d = 2, and the pair is
quadraticbyTheorem4.4. Supposen≥2. ThenthelinearspanI =he ,...,e iisasubsetofU
k+1 n
andanidealinR. As we know,the polynomialsF(z ,...,z )fortwopairs(R,U)and(R/I,U/I)
1 k
coincide. So the assumption is applicable if I 6= 0. There is only one case left, namely, I = 0, or,
equivalently, k = n, md = hei and md+1 = 0. In this situation the element e can be represented