Rational components of Hilbert schemes

Rational Components of Hilbert Schemes

PAOLOLELLA(*) - MARGHERITAROGGERO(*)

ABSTRACT- TheGroÈbner stratum of a monomial ideal jis an affinevariety that parameterizes the family of all ideals havingjas initial ideal (with respect to a fixed term ordering). The GroÈbner strata can be equipped in a natural way with a structure of homogeneous variety and are in a close connection with Hilbert schemes of subschemes in the projective space Pⁿ. Using properties of the GroÈbner strata weprovesomesufficient conditions for therationality of com- ponents ofHilbⁿ_p(z). Weshow for instancethat all thesmooth, irreduciblecom- ponents inHilbⁿ_p(z)(or in its support) and the Reeves and Stillman component HRSare rational. We also obtain sufficient conditions for isomorphisms between strata corresponding to pairs of ideals defining a same subscheme, that can strongly improve an explicit computation of their equations.

1. Introduction.

The aim of the present paper is to investigate effective methods to study the Hilbert scheme of subschemes in the projective spacePⁿ, both on the theoretical and the computational point of view, using GroÈbner basis tools. Several authors have been working in this direction during last years (for instance [3, 21]), but our motivations mainly refer to some ideas and hints contained in the paper [19] by Notari and Spreafico. In order to get a stratification of Hilbⁿ_p(z), they introduce some affine varieties St(j) (here called GroÈbner strata) parameterizing families of ideals in k[X₀;. . .;X_n] having thesameinitial idealjwith respect to a fixed term ordering on the monomials. When only homogeneous ideals ink[X0;. . .;Xn] areconcerned, wewriteSth(j). The ideal defining a GroÈbner stratum springs out from a procedure based on Buchberger's algorithm, but involves a reduction with respect to a set of polynomials which is not a GroÈbner basis.

(*) Indirizzo degli A.: Dipartimento di Matematica dell'UniversitaÁ di Torino, Via Carlo Alberto 10, 10123 Torino, Italy.

E-mail: paolo.lella@unito.it margherita.roggero@unito.it

Supported by PRIN (Geometria della varietaÁ algebriche e dei loro spazi di moduli) co-financed by MIUR (2008).

It is not difficult to realize that the support ofSt(j) only depends on the initial data (the term ordering, the ideal j, etc.), but one cannot be be- forehand sure that different choices in the reduction steps always lead to the same ideal. In other words it is not clear if GroÈbner strata are scheme- theoretically well defined. This is a crucial point that is underlined for instance in [21, Section 3]. In fact, if anyone wants to deduce properties of an Hilbert scheme using a GroÈbner stratum, it is necessary to consider carefully the non-reduced structures, because Hilbⁿ_p(z) can havenon-re- duced components (see [12, 14, 18]). A first achievement in this paper is the following result (see Theorem 3.6):

THEOREMA.The ideal definingSt(j)does not depend on the reduction choices.

In fact we exhibit an equivalent, but intrinsic definition for the ideal of St(j), which by the way also allows a great simplification in the procedure for an explicit computation of this ideal.

A meaningful property that all GroÈbner strata enjoy is that they are homogeneous with respect to some non-standard grading. Homogeneous varieties are of a very special type: for instance they can be iso- morphically embedded in the Zariski tangent space at the origin, which is of course the smallest affine space in which such an embedding can be done. Therefore a smooth homogeneous variety has to be isomorphic to an affine space; in fact, the variety has the same dimension as the space in which it can be embedded. Moreover one can obtain directly the ideal of St(j) in the ``minimal embedding'' in the Zariski tangent space, through a preliminary detection of a maximal set of ``eliminable variables'' (we briefly resume this method in § 4). This is a second key point in our work, because one of the main difficulties usually met studying GroÈbner strata (and even more Hilbert schemes) is due to the huge number of variables that their equations involve, even in very simple cases. Besides the ob- vious computational gain, we would like to underline the interesting theoretical outcome of this method: most of our proofs are obtained just using theminimal embedding.

A second useful tool that can simplify the computation of equations defining a stratum is given in Theorem 4.7. Though the strata corre- sponding to ideals that define a same subscheme are in general non-iso- morphic, however we show that two of them give rise to isomorphic strata when they satisfy suitable sufficient condition, so that we can equivalently take into consideration the most convenient one.

THEOREM B. Let j be monomial ideal in k[X0;. . .;Xn] which is saturated and Borel-fixed w.r.t. the order on the set of variables

X0X1. . .Xn.

i) For everym, there is a set of eliminable variables of the ideal de- fining Sth(j_5m), that contains all variables except at most the ones ap- pearing in polynomials F such that LT(F)ˆX^aX_n^{m jaj}, where X^a is a minimal generator ofj.

ii) If X_{n 1} does not appear in anymonomial of degree m‡1 in the monomial basis ofj, thenSt_h(j_{5m 1})' St_h(j_5m).

iii) Especially, if s is the maximal degree of monomials in the mono- mial basis of j containing Xn 1, then St_h(j_{5s 1})' St_h(j_5m) for every ms.

In § 5 and § 6 we investigate more closely the natural connection be- tween the GroÈbner stratumSt_h(j) and theHilbert schemeHilbⁿ_p(z), where p(z) is theHilbert polynomial ofk[X0;. . .;Xn]=j. As for every idealiwhose initial ideal isj, themodulesk[X0;. . .;Xn]=iandk[X0;. . .;Xn]=jsharethe same Hilbert function, there is an obvious set-theoretic inclusion St_h(j) Hilbⁿ_p(z). However it is not a simple task to understand if this in- clusion is an algebraic embedding or not. The paper [19] deals with the same question, but mainly concerns GroÈbner strata of saturated ideals with respect to the term orderingDegRevLex: note that every subschemeZin Pⁿ can be defined by the saturated idealI(Z). In this paper we prefer to consider a slightly different approach, modeled on the classical construc- tion of the Hilbert schemes (see for instance [1, 11]). For everyZ2 Hilbⁿ_p(z) we consider the idealI(Z)5r, whereris theGotzmann number ofp(z). Asr is theworst Castelnuovo-Mumford regularity for all Z2 Hilbⁿ_p(z), the GroÈbner strata (with respect to any term ordering) of monomial ideals generated in degreercoverHilbⁿ_p(z).

Moreover, the subset ofHilbⁿ_p(z)corresponding toSt_h(j_5r) always con- tains theonecorresponding to Sth(j) and theinclusion can bestrict, be- causepoints inSth(j) correspond to ideals defining subschemes inPⁿwith thesameHilbert function as thesubschemeZˆ V(j), whilebeing in St_h(j_5r) only requires the same Hilbert polynomial. An interesting ex- ample of this type is that of the lexicographic saturated idealLsuch that

k[X₀;. . .;X_n]=Lhas Hilbert polynomialp(z) and whose regularity is indeed

theGotzmann numberrofp(z): in § 7 weshow thatSth(L5r) is isomorphic to an open subset of the Reeves and Stillman componentH_RSof Hilbⁿ_p(z), while in generalSt_h(L) corresponds to a locally closed subscheme of lower dimension (see [22, Remark 4.8]).

The main reason of our setting is contained in Theorem 6.3. Letp(z) be any admissibleHilbert polynomial inPⁿwith Gotzmann numberrand let be a fixed term ordering on monomials ofk[X0;. . .;Xn]. Following the classical construction, weconsiderHilbⁿ_p(z) as a subscheme of a projective spacethrough thePluÈcker embedding of the grassmannianG(t;M), where

Mˆdim_k(k[X₀;. . .;X_n]_r) and tˆM p(r). Thesimpleremark that the

PluÈcker coordinates correspond to sets oftdistinct monomials of degreer (that we can write in decreasing order with respect to), allows us to get a lexicographic total order on them. Ifj₀ is a monomial ideal generated in degree r such that k[X₀;. . .;X_n]=j₀ has Hilbert polynomial p(z) with Gotzmann numberr, then we show that St_h(j₀) is thelocally closed sub- scheme of Hilbⁿ_p(z) given by theconditions that thePluÈcker coordinate corresponding to the monomial basis ofj₀does not vanish and the bigger ones vanish.

As a consequenceweareableto provethat every irreducibleand reduced component of Hilbⁿ_p(z) (or of its support) has an open subset which is a homogeneous affine variety (with respect to a non-standard grading). Especially, if j is generated by the t largest degree r monomials (wecall it a (r;)-segment ideal), then St_h(j) is naturally isomorphic to an open subset of Hilbⁿ_p(z). Therefore we can easily de- duce a few interesting properties of rationality for the components of Hilbert schemes (see Theorem 6.3 iii), Corollary 6.9, Corollary 6.10, Corollary 7.1):

THEOREMC. Let H be an irreducible component ofHilbⁿ_p(z).

If H is smooth, then it is rational. The same holds for its support SuppH.

If H contains a smooth point which corresponds to a(r;)-segment ideal (whereis anyterm ordering), then H is rational. The same holds forSuppH.

The Reeves and Stillman component H_RS ofHilbⁿ_p(z)is rational.

The last item can be obtained as a direct consequence of the previous one, because the lexicographic saturated idealLcorresponds to a smooth point in H_RS, as proved by Reeves and Stillman in [20], and L5r is a (r;Lex)-segment ideal. However, we can also get a new proof of this fact, not applying the quoted result by Reeves and Stillman, but proving that St_h(L_5r) is isomorphic to an affine space using our method based on the minimal embedding (see Theorem 7.3).

In § 8 we present a pseudo-code description of the procedures based on our results, that can be implemented using one of the several softwares for symbolic computation. With such a procedureweareableto writeequa- tions for someGroÈbner strata corresponding to the Hilbert schemeHilb³_4z. In this way wefind a computational confirmation of theresults obtained by Gotzmann in [9], namely thatHilb³_4zhas two components of dimensions 23 and 16 respectively and also some improvements. In fact, we also show that both components of that Hilbert schemearerational (becauseeach of them contains a smooth point corresponding to a segment ideal), they have transversal intersection (studied using a third segment ideal) and finally that the forth segment ideal allowed by the Hilbert polynomial p(z)ˆ4zis singular point whose stratum has dimension 23 and embed- ding dimension 27.

The paper is organized as follows. § 2 contains some general notation.

In § 3, wetakeup theconstruction of GroÈbner strata made in [22] and prove that they are well defined (Theorem 3.6).

In § 4 we discuss the main properties of GroÈbner strata as homo- geneous varieties with respect to a non-standard grading and we obtain some useful criterion in order to know when Borel-fixed monomial ideals with thesamesaturation definethesameGroÈbner stratum (Theorem 4.7).

In§ 5, we focus ourattentiononidealsgenerated indegreer,whereris the Gotzmann number of their Hilbert polynomials, and prove that their GroÈbner strata can bedefined by minors of suitablematrices (Proposition 5.5).

§ 6 represents the heart of the work. We show that there is a close connection between the above quoted matrices defining GroÈbner strata and those appearing in the classical construction of Hilbert schemes and obtain as a consequence the main results of the paper about rational components (Theorem 6.3iii)).

Finally, in § 7, we prove the rationality of the Reeves-Stillman com- ponentH_RSofHilbⁿ_p(z)using our method, based on the minimal embedding and in § 8 we apply this same method in order to perform some explicit computations aboutHilb³_4z.

2. Notation.

Throughout the paper, we will consider the following general notation.

1. During theconstruction of GroÈbner strata, we work on a fieldkof any characteristic, whereas when we study the Hilbert scheme, we will supposethatkis algebraically closed.

2. k[X0;. . .;Xn] is the polynomial ring in the set of variablesX0;. . .;Xn

that wewill often denoteby thecompact notation X, so that

k[X]:ˆk[X0;. . .;Xn]; wewill denotebyX^athe generic monomial ink[X],

where a represents a multi-index (a₀;. . .;a_n), that is X^a:ˆX₀^a0 X_n^an. j will bea monomial ideal in k[X] with basisfX^g1;. . .;X^gtg and Syz(j) its k[X]-moduleof syzygies.

X^a jX^g means that X^a dividesX^g, that is there exists a monomialX^b such that X^aX^b ˆX^g. If such monomial does not exist, we will write X^aj⁴X^g.

will be a fixed term ordering on the set T_X of monomials in k[X]

and wealways assumethatX₀ . . .X_n. As theterm orderis fixed, weoften omit to indicateit. Given a monomialX^a, we re fe r with min (X^a) as thesmallest variabledividing themonomial, that is min (X^a)ˆ minfX_is:t:X_ijX^ag. Wewill denotealso by its extension to the mul- tiplicativegroup of Laurent monomials T_X and thecorresponding total ordering onZ^n‡1 given byab,X^aX^b.

For every polynomialF ink[X] (ork[X;C],k[C]), LT(F) is its leading term with respect to the fixed term ordering; in the same way, ifais an ideal,LT(a) is its initial ideal.

3. We will introduce a second set of variables C_ia that wewill denote withC. Sok[X;C] will bethepolynomial ring in thevariablesXandCand TX;Cthe corresponding set of monomials. The term ordering onTX;Cwill be induced by the term ordering onTXand it will be an elimination ordering of thevariablesXthat will coincidewithonT_X: so wewill denoteby the samesymbolalso this term orderings onT_X;Cand its restriction toT_C.

4. Let G beany polynomial in k[C;X]. An X-monomial of G is a monomial of TX that appears in G considered as a polynomial in the variablesXwith coefficients in the ringk[C]; theX-coefficientsofGarethe elements ofk[C] that arecoefficients of anX-monomial. Notethat theX- coefficients are polynomials, but not necessary monomials.

5. Given any subschemeZinPⁿ, wewill denoteby SuppZits support and byI(Z) the saturated ideal ink[X] that definesZ. Given any ideala, we will denote byV(a) the affine scheme Spec (k[X]=a).

6. Hilbⁿ_p(z) will denote the Hilbert scheme parameterizing all sub- schemes Z in Pⁿ with Hilbert polynomial p(z). r will betheGotzmann number ofp(z), that is the worst Castelnuovo-Mumford regularity among subschemes parameterized by Hilbⁿ_p(z). When we write that an ideal ik[X] belongs toHilbⁿ_p(z), wewill mean thatiis generated in degreer and that theHilbert polynomial of Projk[X]=iisp(z). By abuseof notation wewill say that any such idealihas Hilbert polynomialp(z) referring to the

Hilbert polynomial of the quotient, even if the real Hilbert polynomial ofi is ^z‡n_n

p(z).

3. The ideal of a GroÈbner Stratum.

Now weintroducetheGroÈbner strata and prove some properties, generalizing definitions and results of the paper [22].

DEFINITION3.1. ThetailofX^g with respect toj(and to the fixed term ordering) is theset of monomials:

T_g ˆ

X^a2T_X X^aX^g; X^a2=j (3:1)

Every idealisuch thatLT(i)ˆjˆ(X^g1;. . .;X^gt) has a reduced GroÈbner basis of thetypeff₁;. . .;f_tgwhere:

fiˆX^gi‡ X

X^a2Tgi

ciaX^a (3:2)

and c_ia2k, c_ia ˆ0 except finitely many of them. It is very natural to parameterize the family of all the idealsiby the coefficientsc_ia; in this way it corresponds to a subset ofk^T, whereTˆT_g1 T_gt.

In many interesting cases, T_gi arefinitesets and so k^T is an affine space: this happens for instance ifjis a zero-dimensional ideal or ifis a suitable term ordering; in other cases, for instance when only homo- geneous ideals are concerned,T can be infinite, but we can restrict our interest to a suitable finite subset. The following definition extends and includes all the previous cases.

DEFINITION3.2. Let us fixTˆ fT1;. . .;TtgwhereT_iis a finitesubset of thetail ofX^gi with respect toj. Wewill denotebySt(j;T) thefamily of all ideals i in k[X] such that LT(i)ˆj and whosereduced GroÈbner basis f₁;. . .;f_tis of thetype:

f_iˆX^gi‡ X

X^a2Ti

ciaX^a: (3:3)

Moreover we will use the following special notation:

i) St(j), if TiˆT_gi (of courseonly if T_gi arefinitesets): St(j) para- meterizes all the idealsisuch thatLT(i)ˆj.

ii) St_h(j), ifT_iis thesubset ofT_giof themonomials with thesamedegree asX^gi:St_h(j) parameterizes all the homogeneous idealsisuch thatLT(i)ˆj.

REMARK3.3. It will be clear later that the term ordering affects the construction of a GroÈbner stratum only because it states which monomials can belong to the tails; in fact two different term orderings giving the same tails will lead to the same GroÈbner strata.

Every idealiin thefamilySt(j;T) is uniquely determined by a point in theaffinespaceA^N (NˆP

ijTij) where we fix coordinates Cia corre- sponding to the coefficients c_ia that appear in (3.3). The subset of A^N corresponding to St(j;T) turns out to be a closed algebraic set. More precisely, we will see how it can be endowed in a very natural way with a structure of affine subscheme, possibly reducible or non reduced, that is we will see that it can be obtained as the subscheme ofA^Ndefined by an ideal h(j;T) ink[C], whereCis the set of variablesC_ia.

In the following, we refer to the terminology introduced in Notation 4 for what concerns the polynomials ink[X;C].

DEFINITION3.4. Wewill denotebyh(j;T) andL(j;T) respectively any ideal ink[C] that can beobtained in thefollowing way.

LetB ˆ fF₁;. . .;F_tgbetheset of polynomials ink[X;C] given by:

FiˆX^gi‡ X

X^a2Ti

CiaX^a: (3:4)

Consider any term order ink[X;C] which is an elimination order for thevariablesXand that coincides withfor monomials inT_X; there will beno confusion if wedenoteit by thesamesymbol. With respect to such a term order, the leading term ofF_iisX^gi.

Fix thesubsetPoff(i;j)j14i5j4mgcorresponding to any set of generators for Syz(j);

For every (i;j)2P, le t R_ij be a complete reduction of the S-poly- nomialS(Fi;Fj) with respect toB.

For every (i;j)2P, le tM_ijbe a complete reduction ofS(F_i;F_j) with respect toj.

h(j;T) is theideal in k[C] generated by the X-coefficients of the polynomialsR_ij, (i;j)2P.

L(j;T) is thek-vector space inhCigenerated by theX-coefficients of Mij, (i;j)2P.

It is almost evident, that the definition of h(j;T) is nothing else than Buchberger's characterization of GroÈbner basis if we think to theC_ia's as constant inkinstead of variables. In fact the variablesCdo not appear in

the leading terms of Fi and so their specialization in k commutes with reduction with respect to B. Thus (. . .;cia;. . .) is a closed point in the support of V(h(j;T)) in A^N if and only if it corresponds to polynomials

f₁;. . .;f_mink[X] that area GroÈbner basis. Then the support ofV(h(j;T)) is

uniquely defined; however a priori the ideal h(i;T) could depend on the choices we perform computing it, that is on the choice of the set P of generators for Syz(j) and on thechoiceof a reduction for theS-polynomials S(F_i;F_j) with respect toB(which in general is not uniquely determined).

Thanks again to Buchberger's criterion, we can prove that in facth(j;T) only depends on j, T and of course because it can be defined in an equivalent intrinsic way.

PROPOSITION3.5. Letjk[X],B ˆ fF1;. . .;Ftg k[X;C]andbe as above and consider an idealain k[C]with GroÈbner basisA. The following are equivalent:

i)B [ Ais a GroÈbner basis in k[X;C];

ii) a contains the X-coefficients of all the polynomials in the ideal

(F1;. . .;Ft)k[X;C]that are reduced moduloj;

iii) a contains all the X-coefficients of everycomplete reduction of S(F_i;F_j)with respect toBfor everyi;j;

iv) acontains all the X-coefficients of some (even partial) reduction with respect toBof S(Fi;Fj)for everyi;j;

v) a contains all the X-coefficients of some (even partial) reduction with respect to B of S(F_i;F_j), for every (i;j) corresponding to a set of generators ofSyz(j).

PROOF. i† )ii†: le tGa polynomial in (F1;. . .;Ft)k[X;C] which is re- duced moduloj. By hypothesis,Gmust bereducibleto 0 throughB [ A, so that the next step of reduction have to be performed just usingA. But any step of reduction through Adoes not change theX-monomials and only modifies theX-coefficients; thenG !^A 0, that is everyX-coefficient inG can bereduced to 0 using A: this shows that all the X-coefficients in G belong toa.

ii† )iii†,iii† )iv†andiv† )v†areobvious.

v† )i†: wecan check thatB [ A is a GroÈbner basis using the refined Buchberger criterion (see for instance [4, Theorem 9, pag. 104]). If A ˆ fa1;. . .;arg, a set of generators for Syz(X^g1;. . .;X^gt;LT(a1);. . .;LT(ar)) can be obtained as the union of a set of generators for Syz(X^g1;. . .;X^gt), a set of generators for Syz(LT(a₁);. . .;LT(a_r)) and theobvious syzygies of (X^gi;LT(a_j)). Then:

S(ai;aj)^B[A!0, sinceAis a GroÈbner basis andA B [ A;

S(ai;Fj)^B[A!0, since the leading terms ofaiandFj arecoprimeand a_i; F_j 2 B [ A;

S(F_i;F_j)^B[A!0 in at least one way, by hypothesis. p There are many idealsa fulfilling the equivalent conditions of Propo- sition 3.5: for instance we can consider the irrelevant maximal ideal ink[C]

or any ideal obtained accordingly with condition iv). Moreover, ifasatisfies thoseconditions anda⁰a, then alsoa⁰does, and if the idealsa_lsatisfy the conditions, then also their intersection T

a_l does. As a consequence of these remarks we obtain the proof of the uniqueness of the idealh(j;T) given by Definition 3.4.

THEOREM3.6. Letjand T as above. Then:

i) h(j;T) is uniquelydefined; in fact h(j;T)ˆT

a, a satisfying the equivalent conditions of Proposition3.5.

ii) L(j;T)is uniquelydefined.

PROOF. i):his oneof theidealsa, because it satisfies conditionv); on theother hand, ifasatisfies conditioniii), then clearlyah.

Forii)it is sufficient to observe that the generators forL(i;T) arethe degree 1 homogeneous components (here ``homogeneous'' is related to the usual grading ofk[C] that is theZ-grading with variables of degree 1) of the generators ofh(i;T) given in its construction (Definition 3.4). p By abuseof notation wewill denoteby thesamesymbol St(j;T) the family of ideals and the subscheme inA^Ngiven by the idealh(i;T). Note that h(i;T) is not always a primeideal and so St(j;T) is not necessarily irreducible nor reduced, as the following trivial example shows.

EXAMPLE3.7. Letjˆ(x²;xy)k[x;y] and beany term ordering.

Let us choose Tˆ

T_x²ˆ ;;Txyˆ fyg and construct theideal of the GroÈbner stratumSt(j;T) according to Definition 3.4:

fF1ˆx²; F2ˆxy‡Cyg;

S12ˆyF1 xF2ˆ Cxy^fF1!^;F2gR12ˆ Cxy‡CF2ˆC²y:

Thenh(j;T)ˆ(C²) that isSt(j;T) is a doublepoint in theaffinespaceA¹.

4. GroÈbner strata are homogeneous varieties.

In this section we will see how every GroÈbner stratum St(j;T) is in a very natural way homogeneous with respect to a suitable non-standard grading onk[C], so that wecan apply theniceproperties typical of this kind of schemes and especially those obtained in [22] and in [8].

For themeaning ofj,k[X],fX^g1;. . .;X^gtgandwe refer to Notation 2 and fork[X;C],fF1;. . .;Ftg,St(j;T),h(i;T) to the previous section.

First of all, we recall the definitions and properties that we will use moreoften.

DEFINITION4.1. Wewill considerk[X;C] andk[C] as graded ring over the totally ordered group (Z^n‡1;‡;) with gradinglgiven byl(X^a)ˆa andl(C_ia)ˆg_i a.

As wewill usealso theusual grading overZwhere all the variables havedegree1, wewill always writeexplicitly thesymbol l when the above defined grading is concerned (so, l-degree l with l2Z^n‡1, l- homogeneous of degree l etc.) and leave the simple terms when the usual grading is involved (so, degree r with r2Z, homogeneous of degree retc.).

PROPOSITION4.2. (See [22, Lemma2.8]) i) The gradinglis positive.

ii)h(j;T)is al-homogeneous ideal.

PROOF. i)Let us observe that all the variables have l-degree higher than that of theconstant 1. In factl(X_i)l(1) becauseis a term ordering andl(C_ia)l(1) because,X^giX^aby definition of tails. As well known, this condition is equivalent to the positivity of the grading (see [15, Chapter 4]).

ii)WeobservethatlonTC is therestriction of thegrading onTX;C. Every monomial that appears inFiis of thetypeCiaX^aand so itsl-degree isl(C_iaX^a)ˆl(C_ia)‡l(X^a)ˆg_i. Thus all thepolynomialsF_iarel-homo- geneous and then also theS-polynomialsS(F_i;F_j) and their reductions are l-homogeneous. Finally, the X-coefficients in any l-homogeneous poly- nomial (which arepolynomials ink[C]) arel-homogeneous. p We now recall some properties ofL(j;T) (see also [22, Proposition 2.4]

and [8, Theorem 3.2]).

PROPOSITION4.3. The linear spaceV(L(j;T))can be naturallyidenti- fied with the Zariski tangent space toSt(j;T)at the origin.

If C⁰⁰C is anysubset of ed :ˆdimV(L(j;T)) variables such that L(j;T) hC⁰⁰i ˆ hCi, then h(j;T)\k[C⁰⁰] defines a l-homogeneous sub- varietyinA^ed isomorphic toSt(h;T).

We may summarize the previous result saying that St(h;T) can be embedded in its Zariski tangent space at the origin. This explains the following terminology.

DEFINITION4.4. The number ed is theembedding dimensionofSt(j;T).

The complementC⁰:ˆCnC⁰⁰is amaximal set of eliminable variablesfor h(j;T).

COROLLARY4.5. In the above notation, the following statements are equivalent:

1. St(j;T)'A^ed; 2. St(j;T)is smooth;

3. the origin is a smooth point forSt(j;T);

4. ed4dimSt(j;T).

Note that in general a maximal set of eliminable variables (and so its complementary) is not uniquely determined. However, if C_ia2L(j;T), thenC_iabelongs to any set of eliminable variables; on the other hand, ifC_ia does not appear in any element ofL(j;T), thenC_iadoes not belong to any set of eliminable variables.

There is an easy criterion that allows us to decide if a variable is eliminable or not.

CRITERION4.6. LetLT(F_i)ˆX^gi,LT(F_j)ˆX^gjand letC_ibbe a variable appearing in the tail of Fi. Using the reduction with respect tojof a l- homogeneous polynomialX^dFi X^hFjwe can see that:

i) if X^d‡b2=j and X^{d‡b h} is not a monomial that appears in Fj, then C_ib 2L(j;T);

ii) if X^d‡b2=jand X^b0 ˆX^{d‡b h}is a monomial that appears in F_j, then C_ib C_jb⁰ 2L(j;T)

Moreover if C_ib C_jb⁰ 2L(j;T), then everymaximal set of eliminable variables must contain at least either one of them.

In most cases the numberNˆ jCjis very big andh(j;T) needs a lot of generators so that finding it explicitly is a very heavy computation. On the contraryL(j;T) is very fast to compute and so we can easily obtain a set of eliminable variablesC⁰; a forgoing knowledge ofC⁰allows a simpler com- putation of theidealh(j;T)\k[CnC⁰] that givesSt(j;T) embedded in the affinespaceof minimal dimensionA^ed.

Furthermore, in many interesting cases we can greatly bring down the number of involved variables thanks to another kind of argument.

THEOREM4.7. Letjk[X₀;. . .;X_n]be a Borel-fixed saturated mono- mial ideal with basis B, m anyinteger and h_m:ˆh(j_5m) the ideal of Sth(j_5m)as in Definition3.4.

i) There is a set of eliminable variables forh_mthat contains all vari- ables except at most the ones appearing in polynomials Fiwhose leading term is either X^g2Bmor X^aX^mn ^jaj, where X^a2B5m.

ii) St_h(j_{5m 1}) is a closed subscheme of St_h(j_5m). More precisely St_h(j_{5m 1})' St_h(j_5m;T) where T contains the complete tail of a mono- mial in the basis ofj_5m if it is not divided byX_n, and a tail containing onlymonomials divided byXnotherwise.

iii) If Xn 1 does not appear in anymonomial of degree m‡1in the monomial basis ofj, thenSt_h(j_{5m 1})' St_h(j_5m).

iv) If X_{n 1}appears in N monomials of degree m‡1in the monomial basis ofj, thenedSt_h(j_5m)5edSt_h(j_{5m 1})‡NM, where M is the number of monomials of the basis ofjof degree smaller than m.

v) Sth(j_{5m 1})6' Sth(j_5m)if and onlyif Xn 1appears in monomials of degree m‡1in the monomial basis ofjandj_{5m 1}6ˆj_5m.

vi) If s is the maximal degree of a monomial divided byX_{n 1} in the monomial basis ofj, thenSt_h(j_{5s 1})' St_h(j_5m)for everym5s.

PROOF. i)Let us consider any monomialX^hin themonomial basis of j_5m which does not belong to B5m and such it that could bewritten as X^hˆX^aX^ewhereX^ais a minimal generator ofjof degreed5mandX^eis a monomial of degree m d, X^e6ˆX_n^{m d}. Then among the polynomialsF_i there are:

FˆX^aX^{m d}_n ‡X C_bX^b; F⁰ˆX^a‡e‡X

C_d⁰X^d:

Wehaveto provethat all thevariablesC⁰ that appear inF⁰can beelimi- nated.

TheS-polynomial ofFandF⁰is:

S(F;F⁰)ˆX_n^pF⁰ X^e0FˆX

C_d⁰X^dX_n^p X

CbX^b‡e0:

No monomialX^dX_n^pin thefirst summand belongs toj_5mbecauseX^d2=jand jis saturated and Borel-fixed. Thus, the linear part of the coefficient of X^dX_n^p in thereduction of this S-polynomial will be either C_d⁰ or C_d⁰ C_b. ThenC⁰is a set of eliminable variables forj_5m.

ii) The first part of this statement is a special case of general facts proved in [11, § 3].

We directly prove the second part (which implies the first one). Here we denote byX^aandX^gthemonomials in thebasis ofj_{5m 1}of degreem 1 andmrespectively, and we set:

Ga:ˆX^a‡X CadX^d Gg:ˆX^g‡X

CghX^h

whereX^dvaries among all monomials of degreem 1 in thetail ofX^aand X^hamong those of the same degree asX^gin its tail. Applying theprocedure described in Definition 3.4 on the set of polynomialsGwedefineSth(j_{5m 1}) by an idealhk[C].

Thebasis ofj_5mis made by monomials of the following three types:

monomialsX^g of degreem, that also belong to the basis ofj_{5m 1}; monomialsX^aX_nsuch thatX^ais any monomial of degreem 1 in the basis ofj_{5m 1};

monomials X^aX_i of degree m such that X^a is as aboveand min (X^a)Xi6ˆXn.

Weset:

Fan:ˆX^aXn‡X

C_adX^dXn

F_ai:ˆX^aX_i‡X

C_ait⁰ X^t jtj ˆm X^tX^aX_i F_g:ˆX^g‡X

C_ghX^h

Notethat weusethesamenames for someof thecoefficients that appears in polynomials F and G, so thatFanˆXnGa and FgˆGg. Applying the procedure described in Definition 3.4 on the set of polynomialsFweobtain an idealh⁰k[C;C⁰] definingSt_h(j_5m;T).

Thanks to i) weknow thatC⁰is a set of eliminable variables forh⁰and so Sth(j_5m;T) is also defined byhˆh⁰\k[C]. The statement follows once we show thathˆh.

In order to eliminate the variables C⁰ we consider every monomial X^aXiˆLM(Fai) and reduce it using the polynomialsG. In this way weobtain a polynomialH_ai2(G)k[X;C] such thatX^aX_i‡H_aiis completely reduced w.r.t.j. Then also X^aX_iX_n‡H_aiX_n‡P

C_ait⁰ X^tX_n (i.e. F_aiX_n‡H_aiX_n) is reduced modulo j and moreover it belongs to (F)k[X;C;C⁰] because X_nG(F)k[X;C;C⁰]. ItsX-coefficients belong toh⁰, because the idealh⁰is generated by theX-coefficient of the polynomials in (F)k[X;C;C⁰]) that are reduced moduloj_{m 1}or moduloj, which is thesame(Proposition 3.5 ii) and Theorem 3.6). The X-coefficients ofF_aiX_n‡H_aiX_n arealso theX-coeffi- cients of F_ai‡H_ai, and are precisely the set of polynomials of the type C⁰_ait f_ait(C) that allow us to eliminate the variablesC⁰. So theelimination of C⁰is obtained simply puttingC_ait⁰ ˆf_ait(C). In this wayFaibecomes Hai

that belongs to (G)k[X;C].

Theidealh, obtained fromh⁰eliminatingC⁰, can also beobtained first eliminating C⁰ and after taking X-coefficients, because the procedure of eliminatingC⁰and that of takingX-coefficients commute. Sohis generated by the X-coefficients of polynomials in (XnGa; Hai;Gg)k[X;C] that are reduced moduloj.

Hencehhbecause (XnGa; H_ai;Gg)k[X;C](G)k[X;C].

On theother hand, X_n(G)k[X;C]ˆ(X_nG_a;X_nG_g)k[X;C]

(X_nG_a;G_g)k[X;C]. Moreover two polynomialsQand X_nQhavethesame X-coefficients and either one is reduced modulojif and only theother is.

Henceweobtain theoppositeinclusionhhand conclude.

iii)Weuseii)and prove that in the present hypothesis, St_h(j_5m)' St_h(j_5m;T), whereTis defined as inii). Following Definition 3.4, we obtain theidealh_m ofSt_h(j_5m) using:

F_an⁰⁰ :ˆX^aXn‡X

CadX^dXn‡X

C_as⁰⁰X^s; Xnj⁴X^s F_ai:ˆX^aX_i‡X

C⁰_aitX^t F_g:ˆX^g‡X

C_ghX^hˆG_g:

NotethatFaiandFgareas inii), but all the degreemmonomials of thetail ofXnX^aappear inF⁰⁰_an, and not only thosedivided byXn.

For every monomial X^a of degree m 1 in thebasis of j_{m 1}, le t us consider theS-polynomial:

S(F⁰⁰_an;F_{an 1})ˆX

C_adX^dX_{n 1}X_n‡X

C_as⁰⁰X^sX_{n 1} X

C⁰_aitX^tX_n: By our hypothesis no monomial appearing in it belongs to j_m. In fact X^sXn 12jif and only if it is a minimal generator ofj, which is excluded by

hypothesis because its degree ism‡1, or it is of thetypeX^aXawithX^a minimal generator of j_m and Xaˆmin (X^sXn 1)ˆXn 1, while X^s2=j_m. Then S(F_an⁰⁰ ;Fan 1) is already reduced with respect to j_m and so its X- coefficients belong to h_m. Especially, as both X^dX_{n 1}X_n and X^tX_n are multipleofX_n, whileX^sX_{n 1}is not, the coefficient ofX^sX_{n 1}is simplyC_as⁰⁰ so that eachC⁰⁰_asbelongs toh_m. Hence we can eliminate all the variablesC⁰⁰, just putting them equal to 0. In this wayF⁰⁰_an becomesFanas in (4.7) and St_h(j_5m)' St_h(j_5m;T), where T is as in ii), and weconcludebecause St_h(j_5m;T)' St_h(j_{5m 1}).

iv) By ii), weknow that edSt_h(j_5m)5edSt_h(j_5m;T)ˆ St_h(j_{5m 1}), where the tails defined inTcontain only monomials divided byX_n. Le t us now consider a monomialX^aamong the generators ofjof degree smaller thanmand a generatorX^g of degreem‡1 divided byXn 1. Computing thestratum St_h(j_5m), in thetail of X^aX^m_n ^jaj there is the monomial X^bˆX^g=X_{n 1}not belonging toT. Let us callDthe coefficient ofX^b, that is

FˆX^aX_n^{m jaj}‡. . .‡DX^b‡. . .:

Thinking about thesyzygies of theidealj, it is e asy to se e that in any S-polynomial,F is surely multiplied by a monomialX^d divided at least by onevariableXi; i5n. Therefore in every S-polynomial themonomial X^bX^dˆ(X^bXi)X^d0 belongs to j because of the Borel-fixed hypothesis, so that it can be reduced. Finally there is no equation involving the variable D, so it is free and it cannot be eliminated. Repeating the reasoning for the M minimal generators of degree smaller than m‡1 and for the N generators divided by X_{n 1} of degree m‡1, weobtain thethesis.

v)straightforward applyingiv).vi)straightforward applyingiii). p With the following examples, we want to underline again the not so crucial role played by term ordering in this construction (Example 4.8) and wewant to show (Example4.9 and Example4.10) that theestimateof growth of the embedding dimension of the stratum introduced in Theorem 4.7iv)is a lower bound.

EXAMPLE 4.8. Let us consider the ideals iˆ(X₀;X₁²;X₁X₂) and jˆi₅₂ˆ(X₀²;X₀X₁;X₀X₂;X₀X₃;X₁²;X₁X₂) in thering k[X₀;X₁;X₂;X₃] and thestrata of theideal j according to two different term orderings:

Sth(j;Lex) andSth(j;DegRevLex). In the first case there are at first 24 new variablesC, whereas in the second case they are 23, so we may guess that thefamily of theideals with initial idealjw.r.t.Lexcould bedifferent from the family of the ideals with initial idealjw.r.t.DegRevLex.

However applying Theorem 4.7, we can see thatSth(j;Lex)' Sth(i;Lex) and Sth(j;DegRevLex)' Sth(i;DegRevLex). Now thetails of the3 monomials that generateiare the same w.r.t. both term orders and then (see Remark 3.3)

St_h(j;Lex)' St_h(i;Lex)ˆ St_h(i;DegRevLex)' St_h(j;DegRevLex):

EXAMPLE 4.9. Let us consider the polynomial ring k[X₀;X₁;X₂;X₃], theidealjˆ(X₀²;X₀X₁;X₀X₂⁴;X⁷₁;X₁⁶X₂²) and any term ordering given by a matrix with first row (23;5;2;1). By the previous theorem we know that

St_h(j)' St_h(j₅₃); St_h(j₅₄)' St_h(j₅₅)' St_h(j₅₆);

St_h(j₅₇)' St_h(j_5m); 8 m58 and

edSt_h(j₅₄)5edSt_h(j)‡2 e dSt_h(j₅₇)5edSt_h(j₅₄)‡3

By a direct computation, we find edSth(j)ˆ46, edSth(j₅₄)ˆ50 and edSt_h(j₅₇)ˆ56.

EXAMPLE4.10. There are at most two possible classes of isomorphism for thestrata St_h(L_5m), where L is a lexicographic ideal: St_h(L) and Sth(L5r 1), whereris the maximal degree of a minimal generator, in fact thevariableXn 1 appears (if it does) only in the generator of degree r.

Calledbthe number of generators of degreer, applying Theorem 4.7iv), wehave

edSt_h(L_{5r 1})5edSt_h(L)‡n b:

If the monomial of maximal degree in the basis does not contain the variableX_{n 1}, wehaveSt_h(L_5m)' St_h(L); 8m.

We conclude this section with a result similar to the one stated in Theorem 4.7 that concerns only the case of homogeneous GroÈbner strata w.r.t.DegRevLex.

PROPOSITION 4.11. Let jbe a Borel-fixed saturated monomial ideal and letbe theDegRevLexterm ordering. Then

St_h(j)' St_h(j_5m); 8 m:

PROOF. The arguments to achieve the proof are very similar to the arguments used in the proof of Theorem 4.7. First of all let us consider the

monomials

FaˆX^a‡X CabX^b

corresponding to the monomial basisB_jofjand theidealh(j)k[C] of the stratumSt_h(j).

In order to computeSt_h(j_5m), wehaveto consider again polynomialsF_a as before ifjaj5m; X^a2B_jand new polynomialsGaesuch thatLT(Gae)ˆ X^a‡e,8X^a2B_j; jaj5m;and8X^eof degreem jaj, especiallyX^aXn^{m jaj}. Then by the definition itself ofDegRevLex, thetail ofX^aX_n^{m jaj} contains exactly the monomials in the tail ofX^amultiplied byX_n^{m jaj}. So wecan write

Gaeˆ X^a‡e‡P

E^e_adX^d; 8 X^e6ˆXn^{m jaj}; X^aXn^{m jaj}‡P

CabX^bXn^{m jaj}ˆXn^{m jaj}Fa; if X^eˆXn^{m jaj}

8<

:

henceh(j_5m)k[C;E] (note that in the present case variablesDdo not appear by construction).

By Theorem 4.7i), weknow that all thevariablesEcan beeliminated.

By the same reasoning used in the proof of Theorem 4.7 ii), theideal hˆh(j_5m)\k[C] contains the X-coefficients of a set of S-polynomials corresponding to a set of theS-polynomials of themonomial basis ofj: so

Sth(j)' Sth(j_5m). p

5. GroÈbner strata and regularity.

In the present and following sectionsk[X],andjˆ(X^g1;. . .;X^gt) will be as in the previous, but from now on we will consider only homogeneous ideals (with respect to the usual grading) and T_i will bethecomplete homogeneous tail of X^gi so that theonly involved strata will bethe homogeneous strataSt_h(j) introduced in Definition 3.2ii). Since every tail is fixed by, we will simply denote ideals defining GroÈbner strata byh(j).

Let p(z) be any admissible Hilbert polynomial for subschemes inPⁿ. Our goal is to show that the Hilbert scheme Hilbⁿ_p(z) can becovered by homogeneous strata of the type St_h(j). In order to prove that, it is con- venient to think ofHilbⁿ_p(z)andSt_h(j) as schemes parameterizing the same kind of objects, namely homogeneous ideals ink[X]; as many ideals define thesamesubschemeZPⁿ, the problem is to select a unique ideal ink[X]

for every subschemeZ. Themost common choiceis to associatetoZthe only homogeneous saturated ideal I(Z) such that ZˆProj…k[X]=I(Z)†;

this point of view is that assumed for instance in [19] and in [22], where homogeneous strata of saturated ideals are considered.

Here we prefer a different approach, that directly calls back to the explicit construction of Hilbert schemes (see for instance [1, 11, 23]).

DEFINITION 5.1. Given an admissible Hilbert polynomial p(z) for subschemes inPⁿ, wewill denotebyrtheGotzmann number ofp(z), that is the worst regularity of saturated ideals defining subschemes in Hilbⁿ_p(z). Moreover we set: M:ˆ ^n‡r_n

, t:ˆM p(r), M1:ˆ ^n‡r‡1_n

and t1:ˆ

M1 p(r‡1).

Macaulay's Theorem states that ris the regularity of the lexsegment ideal with Hilbert polynomial p(z) (for thedefinition and themain prop- erties of regularity and for some consequences, we refer to [10]).

As ZˆProj…k[X]=I(Z)† ˆProj…k[X]=I(Z)_5r†, Z can beuniquely identified by the idealI(Z)_5r, which is generated bytlinearly independent degreerhomogeneous polynomialsF₁;. . .;F_tor, more precisely, by thet- dimensionalk-vector spaceI(Z)_r:Hilbⁿ_p(z) can be realized as a closed sub- scheme in the grassmannian of thet-dimensional vector spaces ink[X]r. A t-dimensional vector space ink[X]_rgives a point inHilbⁿ_p(z)if and only if it generates an idealihavingp(z) as Hilbert polynomial.

NOTATION5.2. From now on,i2 Hilbⁿ_p(z)will mean thatiˆI(Z)_5rfor someclosed subschemeZinPⁿwith Hilbert polynomialp(z). Equivalently wecan say thati2 Hilbⁿ_p(z)if and only ifiis an homogeneous ideal ink[X]

with Hilbert polynomialp(z) (for the meaning of ``Hilbert polynomial ofi'' see Notation 6) which is generated in degreer, whereris theGotzmann number ofp(z).

REMARK 5.3. If i2 Hilbⁿ_p(z), then iis r-regular and it has a free re- solution of thetype:

(5:1) 0!k[X]( r l)^nl!. . .!k[X]( r 1)ⁿ¹!k[X]( r)ⁿ⁰!i!0 ([6, Theorem 1.2]). Then we can find a set of generators for the first sy- zygies Syz(i) in degreer‡1.

If we take into consideration the homogeneous GroÈbner strata St_h(j) and select the monomial idealjinHilbⁿ_p(z), weobtain theintended direct relation between GroÈbner strata and Hilbert schemes.

LEMMA 5.4. If j2 Hilbⁿ_p(z), then (at least set-theoretically) Sth(j) Hilbⁿ_p(z).

PROOF. Le tibeany ideal inSth(j). By hypothesisLT(i)ˆjand theni andjshare the same Hilbert function. Thereforeiis generated in degreer and has Hilbert polynomialp(z) and theni2 Hilbⁿ_p(z). p Now we will see that the set-theoretic inclusions are in fact algebraic maps and that for some ideals they are open injections. The crucial point is that the stratum structure (and so its injection in the Hilbert scheme) depends on the ideal j and not on the the corresponding subscheme ZˆProj (k[X]=j). This is not so surprising becausethechoiceof theideal fixes all the allowed deformations, but we want to stress this issue because in [19] the authors underestimated this fact and they made a wrong choice (proof of Corollary 4.4). In fact thestratum of thesaturated lexicographic idealLwith Hilbert polynomialp(z) is not in general isomorphic to an open subset ofHilbⁿ_p(z) (see [22] and Example 4.10), whereas, as we will see, the stratum of its truncationL⁰ˆL_5ris an open subset of the Reeves-Stillman component ofHilbⁿ_p(z).

Letjbea monomial ideal inHilbⁿ_p(z). As seen in § 3 every idealisuch thatLT(i)ˆjhas a (unique) reduced GroÈbner basisff1;. . .;ftg wheref_i is as in Definition 3.2ii). Not every ideal generated bytpolynomials of such a typehasjas initial ideal. In order to obtain equations forSt_h(j) we consider the coefficients c_ia appearing in the f_i as new variables;

more precisely let Cˆ fCia; iˆ1;. . .;t; X^a 2k[X]rnj_r andX^aX^gig be new variables and considert polynomials ink[X;C] of thefollowing type:

F_iˆX^gi‡ X

X^a2Ti

C_iaX^a (5:2)

whereT_iˆTg_i\k[X]_r(Definition 3.1). Weobtain theidealh(j) ofSt_h(j) collecting theX-coefficients of some complete reduction with respect to

F₁;. . .;F_t of all the S-polynomials S(F_i;F_j), corresponding to a set of

generators for Syz(j) (see Theorem 3.6 and Proposition 3.5 v)).

PROPOSITION5.5. In the above notation, letjbe a monomial ideal in Hilbⁿ_p(z) and let A be the t(n‡1)M1 matrix whose entries are the X- coefficients of X_jF_i, for all jˆ0;. . .;n and iˆ1;. . .;t.

Then the idealh(j)of the homogeneous stratumSt_h(j)is generated by the(t1‡1)(t1‡1)minors of A.

PROOF. By abuseof notation wewritein thesameway a polynomial and therows of its X-coefficients. As in Definition 3.4 we consider a

term order onTX;C which is an elimination order of the variablesXand coincides with the fixed term ordering on TX. It is quite evident by elementary arguments of linear algebra, that the ideal ak[C], gen- erated by all (t₁‡1)(t₁‡1) minors, does not change if we perform somerow reduction onA. Le tP bea set oft₁rows whoseleading terms area basis of j_r‡1. If X_hF_i2 P, then it has the same leading term than= oneinP, sayXkFj; wecan substituteXhFiwithXhFi XkFj. In this way therows not in P become precisely all the S-polynomialsS(F_i;F_j) that haveX-degreer‡1.

At the end of this sequence of row reductions, we can write the matrix as follows:

D E S L

! (5:3)

whereDis at1t1upper-triangular matrix with 1's along the main diag- onal, whoserows correspond to P and whosecolumns correspond to monomials inj_r‡1.

Using rows inP, we now perform a sequence of rows reductions on the following ones, in order to annihilate all the coefficients of monomials in j_r‡1, that is theentries of thesubmatrixS: ifa(C) is the first non-zero entry in a row not inPand its column corresponds to the monomialX^g2j_r‡1, we add to this row a(C)X_kF_j, where X_kF_j2 P and LT(X_kF_j)ˆX^g. This is nothing else than a step of reduction with respect tofF1;. . .;Ftg. At the end of this second turn of rows reductions, we can write the matrix as follows:

D E

0 R

! (5:4)

where the rows in (DjE) are unchanged whereas the rows in (0jR) arethe X-coefficients of complete reductions ofS-polynomials inX-degreer‡1.

Thenais generated by the entries ofRand soah(j).

Wecan seethat this inclusion is in fact an equality taking in mind Remark 5.3 and Proposition 3.5 v): thefirst onesays that Syz(j) is gen- erated in degreer‡1 and thesecond onethat in this caseh(j) is generated by theX-coefficients of complete reductions of theS-polynomialsS(F_i;F_j)

ofX-degreer‡1. p

The following corollary just express in an explicit way two properties contained in the proof of Proposition 5.5.

COROLLARY5.6. In the above notation:

the idealh(j)is generated bythe entries of the submatrix R in(5.4);

the vector space L(j)is generated bythe entries of the submatrix L in (5.3).

As already said in the Remark 3.3, this theorem shows one more time that GroÈbner strata equations are substantially independent of the term ordering, that sets only which monomials can appear in the tailsT_i.

6. GroÈbner strata that are open subsets of an Hilbert scheme.

In the present section we will prove that every homogeneous GroÈbner stratumSt_h(j), wherej2 Hilbⁿ_p(z), can be naturally identified with a locally closed subscheme ofHilbⁿ_p(z)and that it is an open subset ofHilbⁿ_p(z)ifjis generated by the firsttmonomials ink[X] with respect to the fixed term ordering . As a consequence we obtain the main results of the paper about therationality of somecomponents ofHilbⁿ_p(z).

For themeaning of p(z), r, t, M, t1, M1 and i2 Hilbⁿ_p(z) we refer to Definition 5.1 and Notation 5.2.

First of all we recall how equations defining Hilbⁿ_p(z) areusually ob- tained (see for instance [1, 11]). Every ideali2 Hilbⁿ_p(z)is generated by the t-dimensional vector spaceir. On theother hand, thanks to Gotzmann's Persistence Theorem (see for instance [10, Theorem 3.8]), at-dimensional vector space V k[X]_r generates an ideal i2 Hilbⁿ_p(z) if and only if dim_khX₀V;. . .;X_nVi ˆt₁.

ThereforeHilbⁿ_p(z)can bethought as thesubschemeof thegrassmannian G(t;M) defined by the previous condition. Moreover by the PluÈcker em- bedding of the grassmannian in a projective spaceP^q,Hilbⁿ_p(z) becomes a closed subscheme (not necessarily irreducible and reduced) ofP^q.

Here we are not interested in finding explicit equations forHilbⁿ_p(z) in P^q, but only equations defining each open subsetU\ Hilbⁿ_p(z), whereUis the open subset ofG(t;M) given by a non-vanishing PluÈcker coordinate.

DEFINITION6.1. Thinking ofk[X]_ras the vector space generated by its monomials, we can identify every PluÈcker coordinate with a suitable monomial idealjgenerated bytmonomials of degreer. Wewill denoteby U_jandH_jrespectively the open subsets ofG(t;M) and ofHilbⁿ_p(z)where the PluÈcker coordinate corresponding tojdoes not vanish.