A Borel open cover of the Hilbert scheme

0.2 MB·

by  Cristina Bertone

BOREL OPEN COVERING OF HILBERT SCHEMES
MARGHERITAROGGERO
9 Abstract. Letp(t)beanadmissibleHilbertpolynomialinPn ofdegreedandGotzmannnumber
0 r. It is well known that Hilbn can be seen as a closed subscheme of the Grasmannian G(N,s),
p(t)
20 whereN =`n+nr´and s=N−p(r),hence, byPlu¨cker embedding, itbecomes aclosedsubset ofa
suitableprojectivespace. LetusdenotebyBthefinitesetoftheBorelfixedidealsink[X0,...,Xn]

p generated by s monomials of degree r. We associate to every monomial ideal J ∈ B, a Plu¨cker
e coordinatepJ. IfUJ istheopensubsetofG(N,s)givenbypJ 6=0,whichisisomorphictotheaffine
S space As(N−s),then HJ =Hilbnp(t)∩UJ isanopen subsetofHilbnp(t) and then canbeseenas an
affinesubvarietyofAs(N−s). Themainresultsobtainedinthispaperarethefollowing:
1
1 i) HJ 6=∅⇔Proj(k[X0,...,Xn]/J)∈Hilbnp(t);
ii) if non-empty, HJ parametrizes all the homogeneous ideals I such that the set NJ of the

] monomialsnotbelongingto J isabasisofk[X0,...,Xn]/I asak-vectorspace;
G iii) the ideal defining HJ as a subvariety of As(N−s) (i.e. in “local” Plu¨cker coordinates) is
A generated indegree≤d+2;
iv) HJ canbeisomorphicallyprojectedintoalinearsubspaceofAs(N−s)ofdimension≤σ(N−s),
h. whereσ isthenumberofminimalgenerators ofthesaturation Jsat ofJ;
t v) uptochanges ofcoordinates inPn,theopensetsHJ coverHilbnp(t),namely:
a
m Hilbnp(t)= G g(HJ).
g∈GL(n+1)
[ J∈B

1
v 1. Introduction
4
8 LetHilbn betheHilbertschemethatparametrizesthesubschemesinPn withHilbertpolynomial
p(t)
1
p(t). We do not have to underline the importance of Borel ideals in most of the main general results
2
. obtained until now about Hilbert schemes, first and foremost the proof of the connectedness due
9
0 to Robin Hartshorne ([6]). We also quote the paper [13] where Alyson Reeves using Borel ideals
9 determines the “radius ” of Hilbn ; in that paper she posed the following as the first open question:
0 p(t)
v: Is the subset of Borel-fixed ideals on a component of Hilbnp(t) enough to determine the component?
We know that every irreducible component (and every intersection of irreducible components) of
i
X Hilbn containsatleastaBorelpoint(i.e. apointdefinedbyaBorelideal),becauseGalligo’stheorem
p(t)
r
a states that every ideal can be deformed to a Borel ideal by a flat deformation. For this reason it is
natural to take into consideration sets of points of Hilbn that are in some sense “collected around”
p(t)
each Borel point.
To this purpose in this paper we give an explicit construction and investigate the properties of
suitable open subsets H of Hilbn where J is Borel fixed ideal, based on the following algebraic
J p(t)
conditiononquotients: I defines a pointinH if the same setofmonomialsgivesak-vectorbasisfor
J
both the quotient modulo I and the quotient modulo J.
More generally, in §3 we consider for every monomial ideal J in P := k[X ,...,X ] the family
0 n
BSt(J) defined as follows: I ∈BSt(J) if I is an homogeneous ideal in P such that P =I⊕hNJi, as
a k-vector space, where NJ is the sous-escalier of J that is the set of monomials that do not belong
ToLuciana.
0WrittenwiththesupportoftheUniversityMinistryfunds.
0Mathematics SubjectClassification2000: 14C05. Keywords: Hilbertschemes,Borelideals
1
2 MARGHERITAROGGERO
to J. If NJ is a finite set, this construction is clearly close to that of Border bases (see [7] and [14]).
On the other hand, we can easily see, as a consequence of a well kown property of initial ideals, that
BSt(J) contains all the ideals I having J has its initial ideal with respect to some term order (cid:22) on
the monomials. Can we find a caracterization of the points in BSt(J) along the same lines as the
theories of Gr¨obner and border bases?
InthelastyearsseveralauthorshavebeenworkingonthefamilySt (J,(cid:22)),calledGro¨bnerstratum,
h
of all the ideals having the same initial ideal J with respect to a fixed term ordering (cid:22), proving that
St (J,(cid:22)) can be endowed of a structure of algebraic variety. We recall here [12], [15], [2] and mainly
h
[1] by Giuseppa Carr`a-Ferro,which is to our knowledge the first one. Usually the algebraic structure
springs out of a procedure based on Buchberger’s algorithm. However in [8] it is shown that most
arguments involving the term ordering can be skiped out and replaced by some linear algebra tools.
Following the same idea that leads to the construction of St (J,(cid:22)), in §3 we show that every
h
ideal I in BSt(J) contains a unique set of homogeneous polynomials of the type G = {f = Xα+
I α
c Xγ / Xα ∈ G } where G is the monomial basis of J, Xγ ∈ NJ and c ∈ k (c 6= 0 only
αγ J J αγ αγ
P
if deg(Xγ) = deg(Xα)). In the case of a Gr¨obner stratum the set of polynomials G is the reduced
I
Gr¨obner basis of I.
Even though in the present more general situation this idea does not function well, if we assume
that J is a Borel ideal we are able to prove, only by using linear algebra, that the Borel stratum
BSt(J) has a very natural and well defined structure of affine variety (Corollary 3.10).
In§4thepreviouslyobtainedresultsonBSt(J)areappliedtothelocalstudyoftheHilbertscheme
Hilbn . Firstly we choose for each point Z in Hilbn only one ideal defining it, which is not the
p(t) p(t)
most usual saturated ideal I(Z), but its truncation I := I(Z) , where r is the Gotzmann number
≥r
of p(t). Every ideal of this type is generated by s:=N −p(r) forms of degree r (where N := n+r ).
(cid:0) n (cid:1)
In this way we obtain the classical embeddings: Hilbn ֒→G(s,N)֒→PM, where the second one is
p(t)
the Plu¨cker embedding.
We observe that each Plu¨cker coordinate corresponds in a natural way to a monomial ideal J,
generated by s monomials of degree r, namely generated by the linear space J ∈ G(s,N). Then
r
we denote p a Plu¨cker coordinate and by U and H respectively the open subsets of G(s,V) and
J J J
Hilbnp(t) given by pJ 6= 0: it is well known that UJ ∼= As(N−s) and then HJ ֒→ As(N−s) is an affine
variety. The main aim of the present paper is to investigate general features of the affine varieties
H under the additional condition that J is a Borel ideal. This limiting hypothesis on J is of the
J
utmostimportancefortheresultstobeobtainandrepeatedlyappearsintheproofs. However,thanks
to Galligo theorem on generic initial ideals, these special open subsets H are sufficient for covering
J
Hilbn , up to the action of the linear group GL(n+1), that is up to changes of coordinates in Pn.
p(t)
Assuming that J is a Borelideal,we firstprovethat H is empty if and onlyif J does notdefine a
J
pointofHilbn ,whileonthecontrarytheopensubsetH ,asanaffinevariety,isnaturallyisomorphic
p(t) J
to the Borel stratum BSt(J) (Corollary 4.3 and Theorem 5.1).
Moreover,weare ableto provethatthe idealI defining H as asubvarietyin As(N−s) (thatis in
J J
“localPlu¨ckercoordinates”pa/pJ),isgeneratedinaratherlowdegree. Moreprecisely,IJ isgenerated
indegree≤d+2,dbeingthedimensionofthesubschemesparametrizedbyHilbn ,thatisthedegree
p(t)
of the Hilbert polynomial p(t). For instance, in the case of points, that is if Hilbn parametrizes
p(t)
0-dimensionalschemes,thenI isgeneratedindegrees≤2. Asimilarresultinthe0-dimensionalcase
J
has been presented by M.E. Alonso, J. Brachat and B. Mourrain at MEGA 2009.
ThoughasetofgeneratorsfortheidealI ⊂k[C](where|C|=s(N−s))canbeobtainedinavery
J
simplewayjustbyusinglinearalgebra(theyaresomeofthe minorsofasuitablematrix: see(5))and
BOREL OPEN COVERING OF HILBERT SCHEMES 3
theirdegreesareresonablylow,thenumbers(N−s)oftheinvolvedvariablesisingeneralhigh. Inthe
paper[8]ananalogoussituationhasbeenanalizedfortheGr¨obnerstratumSt (J,(cid:22)): ifJ isgenerated
h
by the s highest degree r monomials with respect to (cid:22), then Sth(J,(cid:22)) ∼= UJ. Under that stronger
hypothesis, I turns out to be homogeneous with respect to a non-standard positive graduation on
J
k[C]. This allows an embedding of H in its Zariski tangent space at the point corresponding to J
J
itselfandthe projectioncanbe simply obtainedbyeliminating some variablesappearinginthe linear
part of polynomials in I (see [15] and [2]). Simply by assuming that J is Borel, we may define a
J
graduation on k[C], in the same way as in the previous special case, and, again, I turns out to be
J
homogeneous, but the graduation is not always positive.
HoweverwecandefineapartialorderingonasuitablesubsetC′′ ofthevariablesC that“partially”
makesupforthepossiblenon-positivityofthegraduation. Inthiswaywecanprovethatthevariables
C′′ canbeeliminated. ThereforeH becamesasubvarietyinthe affinespaceAσ(N−s),whereσ isthe
J
number of minimal generators for the saturated ideal Jsat and it is often far lower than the number
s of generators of J (Corollary 5.5).
2. Notation
Throughout the paper, we will consider the following general notation.
We work on a algebraically closed ground field k of characteristic 0. P = k[X ,...,X ] is the
0 n
polynomial ring in the set of variables X ,...,X that we will often denote by the compact notation
0 n
X. We will denote by Xα the generic monomial in P, where α represents a multi-index (α ,...,α ),
0 n
that is Xα =Xα0···Xαn. Xα |Xγ means that Xα divides Xγ, that is there exists a monomial Xβ
0 n
such that Xα·Xβ =Xγ. If such monomial does not exist, we will write Xα ∤Xγ.
For every monomial Xα 6= 1 we set min(Xα) := min{i : X |Xα}. If a is a monomial ideal in P,
i
Ga will denote its monomials basis and Na its sous-escalier that is the set of monomials that do not
belong to a. We will denote by a and Na respectively the elements of degree m in either set.
m m
We fix the following order on the set of variables X < X < ··· < X . A monomial Xβ can
0 1 n
be obtained by a monomial Xα through an elementary move if XβX = XαX for some variables
i j
X 6=X or equivalently if there is a monomial Xδ such that Xα =XδX and Xβ =XδX . We will
i j i j
denote an elementary move by Xα → Xβ if X > X and by Xα ← Xβ if X < X . The transitive
i j i j
closure of the elementary moves gives a quasi-order on the set of monomials of any fixed degree that
we will denote by ≻ :
B
Xα ≻B Xβ ⇐⇒ Xα →Xγ1 →···→Xγt →Xβ for some monomials Xγ1,...,Xγt.
An ideal b ⊂ k[X ,...,X ] is Borel-fixed if it is fixed under the action of the Borel subgroup of
0 n
lower-triangular invertible matrices. We will use the following explicit characterization: a monomial
ideal b is Borel fixed if and only if: Xα ∈b and Xβ →Xα =⇒ Xβ ∈b.
Notethat≺ agreewitheverytermordering≺onP suchthatX ≺···≺X ,thatis: Xα ≺ Xβ
B 0 n B
⇒ Xα ≺Xβ.
3. The family of a Borel ideal
In this initial section we want to generalize the construction of the homogeneous Gr¨obner stratum
St (J,(cid:22)) of a monomial ideal J in P (see [1], [12], [15], [2], [8]). Roughly speaking, St (J,(cid:22)) is an
h h
affine variety that parametrizes all the homogeneous ideals I ⊂ P whose initial ideal with respect
to a fixed term ordering (cid:22) is J. The main tools used in the construction of this family are reduced
Gr¨obner bases and Buchberger algorithm. If G is the monomial basis of J, an ideal I belongs to
J
St (J,(cid:22))ifitsreducedGr¨obnerbasisisofthetype{f =Xα− c Xγ /Xα ∈G }wherec ∈k,
h α αγ J αγ
P
4 MARGHERITAROGGERO
Xγ ∈NJ and Xγ ≺Xα (of the same degree). Due to the good properties of Gr¨obner bases, we can
also observe that NJ is a basis of P/I as a k-vector space.
We can now startfrom this last property and try to generalizethe above construction, not consid-
ering any term ordering in P. More explicitly, we want to study the family of all the homogeneous
ideals I ⊂P such that NJ is a basis for the k-vector space P/I. Every such ideal I has a special set
of generators that looks like a reduced Gr¨obner basis. This remark motivates the following:
Definition 3.1. Let J be a monomial ideal in P with monomial basis G . We will call J-local set
J
any set of homogeneous polynomials of the type:
(1) G= f =Xα+ c Xγ / Xα ∈G with Xγ ∈NJ and c ∈k.
n α X αγ Jo αγ
Moreover we will say that G is a J-local basis if NJ is a basis of P/(G) as a k-vector space.
It is quite obvious from the definition, that the ideal generated by a J-local basis has the same
Hilbert function than J.
The following examples shows that not every J-local set is also a J-local basis and that, more
generally, J-local sets or even J-local bases do not have the good properties of Gr¨obner bases.
Example 3.2. In k[x,y,z] let J = (xy,z2) and I = (f = xy +yz,f = z2 +xz). The ideal I is
1 2
generated by a J-local set. However J defines a 0-dimensional subscheme in P2, while I defines a
1-dimensional subscheme (because it contains the line x+z =0). Therefore, {f ,f } is not a J-local
1 2
basis because I and J do not have the same Hilbert function.
Even when the ideal I is generated by a J-local set and they share the same Hilbert function, the
J-localsetisnotnecessarilyaJ-localbasis,thatisNJ isnotnecessarilyabasisforP/I asak-vector
space.
Example 3.3. In k[x,y,z], let J =(xy,z2) and let I be the ideal generated by the J-local set {g =
1
xy+x2−yz , g =z2+y2−xz}. It is easy to verify that both J and I are complete intersections of
2
two quadrics and then they have the same Hilbert function. However, NJ is not a basis for P/I: in
fact zg +yg =x2z+y3 ∈I is a sum of monomials in NJ, hence {g ,g } is not a J-local basis.
1 2 1 2
Lemma 3.4. Let I and J be ideals in P, with I homogeneous and J monomial.
If NJ is a basis for P/I as a k-vector space, then I contains a unique J-local set G .
I
Proof. Let Xα be any monomial in the monomial basis G of J. By hypothesis, the class modulo I
J
of Xα can be written, in a unique way, as a k-linear combination c Xγ of monomials Xγ ∈NJ.
αγ
P
The difference of the two is then a polynomial f ∈ I of the wanted type: note that f must be
α α
homogeneous because I is homogeneous and NJ is a basis of the quotient.
The polynomials f form the only J-local set G contained in I (cid:4)
α I
WhenthemonomialidealJ istheinitialidealofI withrespecttoatermordering,thenthereduced
Gr¨obner basis of I is exactly the J-local set contained in I and it is also J-local basis for I. Hence I
has a J-local basis which acts also as a set of generators. However in general the hypotheses of the
previous lemma do notimply that the J-localset G containedin I is alsoa J-localbasis;in fact the
I
ideal generated by G can be strictly smaller than I.
I
Example 3.5. In k[x,y,z] let J = (xy,z2) and I = (f = xy+yz,f = z2+xz,f = xyz). Both
1 2 3
I and J define 0 dimensional subschemes in P2 of degree 4. In order to verify that NJ is a basis
for k[x,y,z]/I it is sufficient to show that, for every m ≥ 2, the k-vector space V = I +NJ =
m m m
I +hxm,ym,xm−1z,ym−1ziisequaltok[x,y,z] . Form=2,thisisaobvious. Then,assumem≥3.
m m
BOREL OPEN COVERING OF HILBERT SCHEMES 5
First of all we observe that yz2 =zf −f ∈I: then V contains all the monomials ym−izi. Moreover
1 3 m
x2y =xf −f ∈I and xym−1 =ym−2f −zym−1 ∈V : then V contains all the monomials xm−iyi.
1 3 1 m m
Finally, we can see by induction on i that all the monomials xizm−i belongs to V . In fact as already
m
proved, zm ∈V , hence xi−1zm−i+1 ∈V implies xizm−i =xi−1zm−i−1f −xi−1zm−i+1 ∈V .
m m 2 m
However the J-local set G = {f ,f } is not a J-local basis because it does not generate I (see
I 1 2
Example 3.2).
We can recover most of the good properties of Gr¨obner bases in J-local bases, when J is a Borel
fixed monomial ideal.
Theorem 3.6. Let J be a Borel fixed monomial ideal in P with monomial basis G and let I be a
J
homogeneous ideal generated by a J-local set G ={f =Xα+ c Xγ / Xα ∈G }.
I α αγ J
P
Then NJ generate P/I as a k-vector space, so that dim I ≥dim J for every m≥0.
k m k m
Moreover: G is a J-local basis ⇐⇒ I and J share the same Hilbert function.
I
Proof. We will prove, by induction on m, that NJ generates (P/I) namely that every monomial
m m
in J can be written modulo I as a sum of monomials in NJ . We will use every polynomial f as
m m α
a “rewriting low”: in fact, modulo I, every Xα ∈G can be replaced by − c Xγ with Xγ ∈NJ.
J αγ
P
If m = 0 there is nothing to prove. Lt us assume that m > 0 and that the thesis holds in degree
m−1. Consider a monomial Xβ ∈ J . If Xβ ∈ G , we can immediately conclude using f ∈ G .
m J β I
On the contrary, Xβ = XαX for some Xα ∈ J : by the inductive hypothesis, we can rewrite
i m−1
Xα modulo I by − c Xγ which is a sum of monomials of NJ . Then we can rewite Xβ by
αγ m−1
P
− c XγX . ThisisnotalwaysasumofmonomialsinNJ ,becausesomeofthemonomialsXγX
αγ i m i
P
couldbelong to J . IncaseXγX ∈G , weproceedasbefore rewritingit throughthe corresponding
m i J
element in G . If, on the other hand, XγX = XηX with Xη ∈J , we can rewrite Xη using the
I i j m−1
inductive hypothesis. Again, some monomial of NJ multiplied by X could belong to J and so
m−1 j m
on.
This procedure will stop in a finite number of steps because J is Borel fixed. In fact, in this
hypothesis, XγX =XηX , with Xγ ∈NJ , Xη ∈J implies Xη →Xγ, hence X >X . (cid:4)
i j m−1 m−1 i j
If the ideal J in not Borel fixed, the procedure of reduction described in the proof of the previous
result could give rise to a “loop” without end.
Example 3.7. Let us consider the ideals of Example 3.2. The monomial xyz can be reduced modulo
I using f as xyz−zf = −yz2. This last monomial also belongs to J and can be reduced modulo I
1 1
using f as −yz2+yf =xyz, which is again the initial monomial.
2 2
Definition 3.8. For every Borel fixed ideal J in P, the Borel stratum of J is the family, that we
will denote by BSt(J), of all homogeneous ideals I such that NJ is a basis of the quotient P/I as a
k-vector space.
We can reformulate the previous results in the following way:
Corollary 3.9. Let J be a Borel ideal. Then:
I ∈BSt(J) ⇐⇒ I is generated by a J-local basis.
Since a J-local basis for an ideal I is uniquely determined, we can give an explicit description of
the family BSt(J) using this special set of generators.
Let us consider a new variable C for every monomial Xα in the monomial basis G of J and
αγ J
for every monomial Xγ ∈ NJ of the same degree as Xα. We can denote by C the set of these new
variables.
6 MARGHERITAROGGERO
Moreover, let us consider the polynomials F = Xα+ C Xγ ∈ k[X,C] and collect them in a
α αγ
P
set:
(2) G ={F =Xα+ C Xγ / Xα ∈G }.
α X αγ J
We can obtain every ideal I ∈BSt(J) specializing (in a unique way) the variables C in k, but not
αγ
every specialization gives rise to an ideal in BSt(J).
Corollary 3.10. If J is a Borel fixed ideal in P, the Borel stratum BSt(J) is an affine subvariety in
AK, where K =#(C), that contains the origin, the point corresponding to J itself.
Proof. Thanks to Theorem 3.6, a specialization C 7→ c ∈ k transforms G in a J-local basis G if and
only if dim ((G) ) ≤ dim (J ) for every m ≥ 0. We can then consider for each m the matrix A
k m k m m
whose columns correspond to the degree m monomials in P and whose rows contain the coefficients
of those monomials in every polynomial of the type XγF such that deg(Xγ+α)=m: every entry in
α
A is either 0, 1 or one of the variables C. The ideal of BSt(J) in k[C] is then generated by all the
m
minors in A of order dim (J )+1 for every m≥0. (cid:4)
m k m
Remark 3.11. Due to Gotzmann’s persistence, in order to obtain a set of generators for the ideal of
BSt(J) it will be sufficient to take into consideration the matrices A only for m≤r+1, where r is
m
the Gotzmann number of the Hilbert polynomial of P/J.
4. Borel strata and Hilbert schemes
In this section we will apply the results obtained in the previous one and prove that Borel strata
give in a very natural way an open covering for every Hilbert scheme of subschemes of Pn.
First of all, let us recall how Hilbert schemes can be constructed and introduce some related
notation. Inthefollowing,p(t)willbeanadmissibleHilbertpolynomialinPn ofdegreedandHilbn
p(t)
will denote the Hilbert scheme parameterizing all the subschemes Z in Pn with Hilbert polynomial
p(t). WewillalwaysdenotebyrtheGotzmannnumberofp(t),thatistheworstCastelnuovo-Mumford
regularityforthesubschemesparametrizedbyHilbn ,byN the number n+r ,andbysthe number
p(t) (cid:0) n (cid:1)
N −p(r).
Every subscheme Z ∈Hilbn can be obtained by using many different ideals in P, first of all the
p(t)
saturated ideal I(Z) = ⊕ H0I (i), whose regularity is lower than or equal to r. Here we prefer to
i Z
consider the “truncated” ideal I = I(Z) , which is generated by s linearly independent forms of
≥r
degree r and so it is uniquely determined by a linear subspace of dimension s in the k-vector space
V = P of dimension N. Thus, Hilbn can be embedded in the Grassmannian G(s,V) of the s-
r p(t)
dimensionalsubspacesofV. Byabuseofnotation,wewillwriteI ∈G(s,V)ifI istheidealgenerated
byI ∈G(s,V)andalsoI ∈Hilbn ifmoreoverI definesasubschemeinPn withHilbertpolynomial
r p(t)
p(t).
Using Plu¨ckercoordinates,G(s,V) becames a closedsubvarietyof the projective space PM, where
M+1= N : eachPlu¨ckercoordinatecorrespondsto a set ofs differentelements in a givenbasis for
(cid:0)s(cid:1)
V: the most natural basis to choose from is the one given by all the degree r monomials.
Inthe following,wewilldenotebyQthesetofthemonomialidealsinG(s,V)andbyB thosethat
are Borel fixed.
There is an obvious 1−1 correspondence between Plu¨cker coordinates and ideals in Q: then we
will denote by p the Plu¨cker coordinate corresponding to J ∈Q.
J
If I ∈ G(s,V), due to Gotzmann’s persistence we can say that I ∈ Hilbn if and only if
p(t)
dim (I ) ≤ s := n+r+1 − p(r +1). This condition directly gives equations defining Hilbn
k r+1 1 (cid:0) n (cid:1) p(t)
as a subscheme of G(s,V).
BOREL OPEN COVERING OF HILBERT SCHEMES 7
Remark 4.1. A different choice of generators for V corresponds to a linear change of Plu¨cker coor-
dinates, but not to all of them. If for instance we consider e linear change of variables in P and the
basis of V given by the monomials in the new variables, thecorresponding linear change of coordinates
does not modify the equations of G(s,V) in PM, while not every permutation of variables in PM does
the same.
As well known, G(s,V) can be covered by the open subsets U given by p 6= 0, J ∈ Q. Each U
J J J
is isomorphic to the affine space As(N−s).
Since Hilbn ֒→ G(s,V) is in fact a closed embedding, the intersections H = U ∩Hilbn give
p(t) J J p(t)
an open covering of Hilbn by affine varieties. For the aim of the present paper we will consider a
p(t)
slightly different open covering of Hilbn : more precisely we will prove that Hilbn can be covered
p(t) p(t)
by the open subsets H , where J varies among the Borel fixed ideals with Hilbert polynomial p(t),
J
and by the open subsets that we can obtain from these up homogeneouschanges of coordinates in P,
that is under the action of the linear group GL(n+1).
First of all we prove that if J ∈ B, than H is either empty or it is the Borel stratum BSt(J)
J
defined in §3.
Proposition 4.2. Let J an ideal in B and let I be a homogeneous ideal generated by a J-local set G .
I
If J ∈/ Hilbn then also I ∈/ Hilbn .
p(t) p(t)
If, on the other hand, J ∈Hilbn , then the following are equivalent:
p(t)
i) I ∈Hilbn
p(t)
ii) G is a J-local basis;
I
ii) NJ is a basis for (P/I) as a k-vector space;
r+1 r+1
iv) dim I =dim J .
k r+1 k r+1
Proof. The first statement and “i) ⇒ ii)”are straightforward consequence of Theorem 3.6. The
implications “ii) ⇒ iii)” and “iii) ⇒ iv)” are obvious. Finally, “iv) ⇒ i)” follows from Gotzmann
persistence (see Remark 3.11). (cid:4)
We can summarize the previous proposition in terms of the open subsets H as follows:
J
Corollary 4.3. If J ∈B, then:
(1) H =∅ ⇐⇒ J ∈/ Hilbn .
J p(t)
(2) If J ∈Hilbn , then: I ∈H ⇔ I ∈BSt(J).
p(t) J
Proof. Both statements are straightforward consequence of Proposition 4.2 and of the definitions of
H and BSt(J). (cid:4)
J
If J ∈ B, the condition J ∈ Hilbn is sometimes expressed in leterature saying that G is a
p(t) J
Gotzmann set.
Definition 4.4. The Borel covering of Hilbn is the family containing the open subsets of the type
p(t)
H , where J ∈ B ∩Hilbn , and also containing the open subsets that can be obtained from these
J p(t)
through a linear change of coordinates in P, that is of the type g(H ), g ∈GL(n+1).
J
Now we prove that the expression “Borel covering” is correct.
Proposition 4.5. The “Borel covering” of Hilbn is in fact an open covering, that is:
p(t)
Hilbn = g(H ).
p(t) G J
g∈GL(n+1)
J∈B
8 MARGHERITAROGGERO
Proof. Let I ∈ Hilbn be any ideal and (cid:22) any term ordering on monomials in P. Due to Galligo
p(t)
Theorem ([3]), in general coordinates the initial ideal J of I is Borel fixed. Then, as well known, I
andJ havethe same Hilbert polynomial. Since I is generatedindegree r, which is its regularity,also
J is generated in degree r and so J ∈Hilbn ∩B. Then I belongs to the open subset H .
p(t) J
Hence every ideal I ∈Hilbn belongs to some of the open subsets of the Borel covering. (cid:4)
p(t)
5. The ideal defining H in As(N−s)
J
InthepresentsectionwewanttostudythefamilythatparametrizesalltheidealsI ∈Hilbn with
p(t)
non-zero Plu¨cker coordinate p , namely the open set H of Hilbn , and its properties as an affine
J J p(t)
variety. In the previous section it is proved that in the non-trivial case J ∈Hilbn and H contains
p(t) J
the same ideals than the Borel stratum of BSt(J).
In §3 we proved that the Borel stratum BSt(J) can be seen as an affine subscheme in a suitable
affine space AK (see Proposition 3.10) and in §4 that H is a closed subscheme of the open subset
J
U ≃ As(N−s) of the Grassmannian G(s,V), hence H is an affine scheme. Now we will prove that
J J
in fact H and BSt(J) are isomorphic.
J
Theorem 5.1. If J ∈Hilbn ∩B, then H ≃BSt(J) as affine varieties.
p(t) J
MoreovertheidealI definingH asasubschemeofAs(N−s),thatisin“local”Plu¨ckercoordinates,
J J
is generated in degree ≤d+2.
Proof. First of all we verify that the number K of corrdinates C is precisely s(N −s). In fact if we
write the set of homogeneous polynomials of (2) in the present hypothesis:
(3) G ={F =Xα+ C Xγ / Xα ∈G , Xγ ∈NJ }
α X αγ J r
we can see that |C|=|G |·|NJ |=s(N −s).
J r
Moreover, every parameter Cαγ is nothing else than the local Plu¨cker coordinate pa/pJ where a
is the monomial ideal generated by Ga = (GJ ∪{Xγ})\{Xα}. Furthermore, HJ as a subscheme of
As(N−s) and BSt(J) as a subscheme of AK are defined in the same way, that is imposing that the
dimension in degree r+1 be s :=dim (J ) (see Proposition 4.2).
1 k r+1
Forthesecondpartofthethesis,wefollowtheproofofProposition3.10. Letusconsiderthematrix
A = A , whose columns correspond to the degree r+1 monomials in P and whose rows contain
r+1
the corresponding coefficients of the polynomials X F . Every monomial in J can be written, not
i α r+1
necessarily in a unique way, as a product of a monomial Xα ∈ J and a variable X . Let us choose
r i
for each monomial in J one of this expressions X Xα and order the columns of A so that the first
r+1 i
s columns correspond to the chosen forms X Xα for the monomials of J , ordered in decreasing
1 i r+1
order with respect to the index i. Moreover let us order the rows of A so that the first s of them
1
contain the coefficient of X F corresponding to and in the same order as the first s columns.
i α 1
Claim: The square submatrix D of A given by the first s rows and columns has determinant
1
equal to 1. More precisely, it is an upper triangular matrix of the following special form:
I(n) • • • • •
 
0 I(n−1) • • • •
 ... ... ... • • • 
(4) D= 
 0 ... 0 I(k) • • 
 
 .. .. .. .. 
 . . . 0 . • 
 
 0 ... ... ... 0 I(0) 
where each I(k) stands for an identity matrix and corresponds to the group of rows and columns
“X Xα”.
k
BOREL OPEN COVERING OF HILBERT SCHEMES 9
The coefficient of X Xα in X F is 1 and so the elements on the diagonal of D are all 1. All the
i i α
other non-zero entries in D are coefficients of some monomial Xγ ∈ NJ that is variables C such
r αγ
that X Xγ ∈ J, that is X Xγ = X Xβ with Xβ ∈ J . Then Xγ ← Xβ, because J is Borel fixed.
i i j r
Hence X >X . This shows that we can find a non-constant entry in the row X F only on columns
i j i α
corresponding to variables X strictly lower than X (i.e. where there is a “•” in the above picture).
j i
This completes the proof of the claim.
The ideal I of H is then generated by the determinants of the submatrices of order s + 1
J J 1
containing D. Before computing those determinants we can row-reduce A with respect to the first s
1
rows. (Note that this is nothing else than the reduction procedure of Theorem 3.6 that uses the F
α
as rewriting lows).
In order to prove that I is generated in degree ≤d+2 we choose to order the rows and columns
J
of A in a special way: for every monomial in J we chose the form X Xα with X = min(X Xα)
r+1 i i i
and Xα ∈ J (note that the hypothesis J Borel fixed allows this special choice). The key point in
r
our argument is that for every Xγ ∈ NJ , the variable X = min(Xγ) is lower than or equal to X ,
r j d
because a Borelideal J candefine a d-dimensionalsubscheme only if Xr ∈J. This implies that we
d+1
can find non-constant entries in A only in columns correspondings to a j-block with j ≤d.
Thus A assumes the following easier form:
I(n,...,d+1) • • • • ... •
 
0 I(d) • • • ... •
 .. .. .. .. .. .. 
 . . . • . . . 
 
(5) A= 0 0 0 I(0) • ... • 
 
 ⋆ • • • • ... • 
 
 .. .. .. .. .. .. .. 
 . . . . . . . 
 
 ⋆ • • • • ... • 
where there is a big identity block I(n,...,d+1) corresponding to the variables X ,...,X and
n d+1
“⋆” stands for a sequence of entries that are all 0 except at most a 1.
Then the ideal I is generated by polynomials of the type:
J
I(n,...,d+1) • • • •
 
0 I(d) • • •
(6) det ... ... ... • ... .
 
 0 0 0 I(0) • 
 
 ⋆ • • • • 
After a partial row reduction, we obtain:
I(d) • • •
 ... ... • ... 
(7) det
 
 0 0 I(0) • 
 
 ◦ ◦ ◦ ◦ 
where “◦” stands for a linear forms in k[C]. It is easy to controll (for instance by a rows reduction of
the last row) that the determinant is a polynomial in k[C] of degree ≤d+2. (cid:4)
The ideal I in k[C] that defines H is then generated in a quite small degree, but it lives in a
J J
polynomialring with a very high number ofvariables. Now we will show that H can be obtained by
J
an ideal in a “smaller” polynomial ring. More precisely, we will prove that H can be isomorphically
J
projected in a linear space which is in general of far lower dimension than s(N −s). This also will
give an upper bound for the dimension of H itself.
J
10 MARGHERITAROGGERO
Definition 5.2. In the same hypothesis of Theorem 5.1, let E be the following set of monomials:
Xβ ∈E ⇐⇒X Xβ =X Xα with X 6=X and Xα ∈J .
0 i i 0 r
Let us denote by C′′ the set of variables C with Xβ ∈E and by C′ its complementary C\C′′.
βγ
In practice, E contains all the monomial of J except those that can be obtained multiplying by a
r
suitable power of X the minimal generators of the ideal b=Jsat. Hence, every monomial in E is a
0
product X Xη with l 6= 0 and Xη ∈ b. Let us split the set of monomials E in the disjoint subsets E
l l
accordingly with the maximal X such that Xα/X belongs to b.
l l
It will be usefull to consider in C′′ the following quasi-order.
Definition 5.3. For every Cβγ,Cβ′γ′ ∈C′′ we put Cβγ ≻Cβ′γ′ if Xβ ∈El, Xβ′ ∈El′ with l>l′ or
l=l′ and Xβ ≻ Xβ′.
B
Theorem 5.4. Let b be a saturated, Borel fixed ideal such that J = b ∈ Hilbn . Then the ideal
≥r p(t)
I ∩ k[C′] of k[C′] defines an affine variety isomorphic to H , that is H can be isomorphically
J J J
projected into the affine space A|C′| given by C′′ =0.
Proof. First of all, we prove that for every variable C ∈ C′′ we can find a polynomial F ∈ I in
βγ βγ J
which C only appears as a degree 1 monomial and that the set of polynomials {F } allows the
βγ βγ
complete elimination of the variables C′′.
Let us consider again the matrix A defined in Theorem 5.1, but now we choose to write every
monomial in J in the special form X Xα where X is the maximal variable such that Xα ∈J .
r+1 i i r
Bythedefinition,foreveryXβ ∈E wehaveX0Xβ =Xi0Xα0 wherei0 >0andXi0Fα0 corresponds
to one of the first s rows in A, that is to one of the rows of D. Thus X F corresponds to a row
1 0 β
which is not in that first group of s . The polynomial F we are looking for is the determinant of
1 βγ
the (s +1)×(s +1) matrix D′ given by the first s rows and columns and by the row of X F and
1 1 1 0 β
the column of X Xγ.
0
If X 6= X , every column in D′ that corresponds to a monomial of the type X Xα contains
n i0 n
only 0 entries, except 1 in the rows X F : then we can exclude those columns and lines without
n α
modifying the determinant. Again, we can do the same for the group of columns X Xα for every
i
i=n−1,...,i0+1 and also for the group of columns corresponding to XiXα except Xi0Xα0. Now
F is the determinant of a matrix M as in the following picture:
βγ
XXi0FFα0α′  10 •1 •• •• •• 
i0+1
...  ... 
 
(8) X Fα  ... 0 1 • • .
i  
 ... ... 
 
...  0 ... 0 1 • 
 
X0Fβ  1 0 ... 0 Cβγ 
It remains to prove that the set of polynomials {F / C ∈C′′} allows the complete elimination
βγ βγ
of the setof variablesC′′. For this we observethat for everyparameterC ∈C′′ appearing as a • in
αδ
M it holds C ≻C with respect to the quasi order of Definition 5.3.
βγ αδ
In fact, as a consequence of the roule we adopted choosing the first rows, Xβ ∈E , while if X F
i0 i α
corresponds to one of the first s rows of M, then Xα ∈ E for some l ≤ i ≤ i . Then C ≻ C if
1 l 0 βγ αδ
i>i0 and also if Xα =Xα0 ∈Ei0 because X0Xβ =Xi0Xα0 implies Xβ →Xα0.
(cid:4)