LOW DEGREE EQUATIONS DEFINING THE HILBERT SCHEME

LOW DEGREE EQUATIONS DEFINING THE HILBERT SCHEME

arXiv:1104.2007v1 [math.AG] 11 Apr 2011

JEROME BRACHAT, PAOLO LELLA, BERNARD MOURRAIN, AND MARGHERITA ROGGERO Abstract. The Hilbert scheme Hilbn p(t) , parametrizing closed subschemes in the projective space Pn with Hilbert polynomial p(t) of degree d, is classically constructed as a subscheme of a suitable grassmannian, so that it becomes a closed projective subscheme through the associated Pl¨ ucker embedding. In this paper we find new global equations for Hilbn p(t) and especially we prove that Hilbn can be defined scheme-theoretically by homogeneous polynomials of degree p(t) smaller than or equal to d + 2 in the Pl¨ ucker coordinates. We obtain these equations using a special set of generators of any exterior power of a vector space, that depend linearly on the Pl¨ ucker coordinates of such vector space in the corresponding grassmannian. Firstly, we exploit these generators to obtain in a simpler way the well-known equations for Hilbn p(t) by Iarrobino and Kleiman and those conjectured by Bayer in 1982 and then proved by Haiman and Sturmfels in 2004. Finally, using combinatorial properties of Borel-fixed ideals and their relations with the geometry of the Hilbert scheme, we obtain our new equations. A procedure for computing this set of equations comes out directly of our proof and it can be fruitfully used in simple cases.

Introduction The study of the Hilbert scheme is a very active domain in algebraic geometry. The Hilbert n scheme HilbPp(t) (k), associated to an admissible Hilbert polynomial p(t) and a projective space Pn = Proj k[x0 , . . . , nn ], was firstly introduced by Grothendieck [9] and parametrizes the set of all subschemes of Pn with Hilbert polynomial p(t). Let S = k[x0 , . . . , xn ] be a polynomial ring in n + 1 variables with coefficients in a field k of characteristic 0 and q(t) = t+n − p(t) be the volume polynomial associated to p(t). Classically n p(r)

the Hilbert scheme is constructed as a subscheme of the grassmannian GrSr (k) (see [9, 16]) of the subspaces of dimension q(r) in the base vector space of dimension n+r of polynomials of n degree r, for an integer r sufficiently large. Two important results by Gotzmann [7], known as Gotzmann’s Regularity Theorem and Gotzmann’s Persistence Theorem, give a method to compute the degree r and the condition to be n p(r) imposed on the points of GrSr (k) in order to define HilbPp(t) (k). The first result allows to compute the minimal suitable degree r directly from the polynomial p(t) and the second theorem p(r) ensures that in order for an ideal Ir ∈ GrSr (k) to have volume polynomial q(t) it is sufficient to check that dim Ir+1 = q(r + 1). Applying these results, in the years many authors dealt with the problem of determining a set of explicit equations defining scheme-theoretically the Hilbert scheme as subscheme of either a grassmannian or a product of two grassmannians or a projective space through the Pl¨ ucker embedding. There are at least a couple of motivations that from our point of view makes interesting the study of this problem. The first one is the following theoretical question, still open: can we determine a set of equations that defines ideal-theoretically the Hilbert n p(r) scheme HilbPp(t) (k) as subscheme of the grassmannian GrSr (k)? 2010 Mathematics Subject Classification. 14C05, 15A75, 13P99. Key words and phrases. Hilbert scheme, Grassmannian, Exterior algebra, Borel-fixed ideal. 1

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J. BRACHAT, P. LELLA, B. MOURRAIN, AND M. ROGGERO

The second motivation is more practical and concerns with projecting effective methods for comn puting a set of equations defining HilbPp(t) (k) scheme-theoretically as subscheme of the grassp(r)

mannian GrSr (k). In fact the number of variables involved in such a computation is very large for the intrinsic complexity of the problem and so it cannot be reduced, therefore we can only try to pull down as much as possible the degree and the number of equations to at least think to do some computation in non-trivial cases. At the moment the results known give equations of quite large degree, even in simple cases. Iarrobino and Kleiman [12, Proposition C.30] determined a set of equations of degree q(r + 1) + 1 and later Bayer [3] conjectured and then Haiman and Sturmfels [10] proved that there exists a set of equations of degree n + 1. These two bounds prove to be too large, even in the simplest 3 non-trivial cases. For instance for the Hilbert scheme HilbP2 (k) of 2 points in P3 , the equations by Iarrobino and Kleiman have degree equal to 19 and those by Bayer, Haiman and Sturmfels degree equal to 4. More recently Alonso, Brachat and Mourrain [1] showed that for Hilbert schemes of points, a set of equations can be found in degree equal to 2. The basic tool used is represented by border basis, which work only on the case of zero-dimension ideals, so that using this construction no improvement can be deduced for non-constant Hilbert polynomials. On the other hand, from a local perspective Bertone, Lella and Roggero [4] proved that n HilbPp(t) (k) can be covered by open subsets defined in local Pl¨ ucker coordinates (that is in affine open subsets of the grassmannian) by equations of degree smaller than or equal to d + 2, where d is the degree of the Hilbert polynomial. These two bounds agree in the case of constant Hilbert polynomials, so we have been encouraged to mix ideas and techniques in order to extend the bound obtained for local equations to global ones. The main result of our paper is the following: Theorem 5.3. Let p(t) be an admissible Hilbert polynomial in Pn of degree d and Gotzmann n p(r) number r. HilbPp(t) (k) can be defined as a closed subscheme of the grassmannian GrSr (k) by equations of degree smaller than or equal to d + 2. Excluding the trivial case of hypersurfaces, this bound is lower than the other two and seems to be more consistent with the object we are studying, because it only depends on the dimension n d of the subschemes parametrized by HilbPp(t) (k). ucker Let us now examine the structure of the paper. In Section 2 we discuss properties of the Pl¨ q coordinates. In particular starting from the coordinates of a point w ∈ P(∧ E) corresponding to a vector subspace V = hv1 , . . . , vq i of a vector space E, that is w = v1 ∧ · · · ∧ vq , we construct a set of generators of ∧l V , for any l = 1, . . . , q, depending linearly on the Pl¨ ucker coordinates of w. Afterward we show that these sets of generators allow us to express in a simpler way the conditions defining the Hilbert scheme as subscheme of the grassmannian. As application of this idea, in Section 3 we give a simple proof of the result by Iarrobino and Kleiman and in Section 4 we obtain in the same simple way the result by Bayer, Haiman and Sturmfels. Indeed both bounds naturally arise applying our construction on the exterior product and the different results only depend on two slightly different strategies. In Section 5, we begin introducing the last ingredient, on which our proof hard relies, that is combinatorial properties of Borel-fixed ideals and their relation with the geometry of the Hilbert scheme. Then we give a constructive proof of the main result of the paper (Theorem 5.3), first on an algebraically closed ground field of characteristic 0 and in the last section over a local ring (Theorem 6.12).


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A procedure for computing this set of global equations comes out directly of the proof, but even if they are much simpler than those introduced by Iarrobino-Kleiman and Bayer-HaimanSturmfels, their computation remains hard to achieve except in a few easy cases. Other approaches, for instance as in [14, 4] by a local perspective, appear to be more promising for an effective study of a specific Hilbert scheme. 1. General setting As already said in the introduction, S will denote the polynomial ring k[x0 , . . . , xn ] with coefficients in a ground field k of characteristic 0. For denoting monomials, we will use the multi-index notation, that is xα0 0 · · · xαnn = xα for every α = (α0 , . . . , αn ) ∈ Nn+1 . Given a homogeneous ideal I ⊂ S, for any t ∈ N, It is the vector space of the polynomials of degree t in I. p(t) and q(t) will be the Hilbert polynomials of the graded modules S/I and I. Following the notation of [7], we will refer to p(t) as Hilbert polynomial of I and to q(t) as volume polynomial of I. Obviously p(t) + q(t) = n+t n . Usually r will be the Gotzmann number of the Hilbert polynomial p(t), that is the highest Castelnuovo-Mumford regularity of the sheaf of ideals associated to a subscheme X ⊂ Pn with Hilbert polynomial p(t). It can be easily computed applying the Gotzmann’s Regularity Theorem [8, Theorem 3.11], indeed writing p(t) in the unique way as t + a0 t + a1 − 1 t + as − s p(t) = + + ... + , a0 > a 1 > . . . > a s , a0 a1 as the Gotzmann number r is equal to s + 1.

Borel-fixed ideals are ideals fixed by the action of the Borel subgroup of the upper triangular matrices of the linear group GL(n + 1). In our assumption on the characteristic of the ground field, they have a strong combinatorial characterization, precisely supposing the variables ordered x as xn > . . . > x0 , a Borel ideal I is a monomial ideal and for any monomial xα ∈ I, xji xα also belongs to I, for all j > i and xi | xα . For further details see [15, 4, 13]. It is well known that the Castelnuovo-Mumford regularity of the ideal sheaf of a subscheme X in Pn coincides with the regularity of the saturated ideal I(X) ⊂ S defining X and that the regularity of a Borel-fixed ideal is equal to the maximal degree of a generator in its monomial basis. So any saturated Borel-fixed ideal I with Hilbert polynomial p(t) is generated in degree smaller than or equal to the Gotzmann number r. In particular the saturated lexsegment ideal associated to the volume polynomial q(t) has regularity equal to r. n

In the following, we will denote by HilbPp(t) (k) the Hilbert scheme parameterizing subschemes of the projective space Pn with Hilbert polynomial p(t). According to the classical construction n of the Hilbert scheme (see [9, 16]), HilbPp(t) (k) can be identified with a subscheme of the grassp(r)

mannian GrSr (k) parametrizing the vector subspaces of dimension q(r) of the vector space Sr . Indeed for any saturated ideal I ⊂ S with Hilbert polynomial p(t), the eqality dimk Ir = q(r) always holds true because the regularity of I is smaller than or equal to the Gotzmann number r. Therefore the key point is to determine the conditions for an ideal J = (Jr ) ⊂ S, generated by a vector subspace Jr ⊂ Sr of dimension q(r), to have volume polynomial q(t). Then we have Jr+1 = hx0 Jr , . . . , xn Jr i by definition and dimk Jr+1 > q(r + 1) by Macaulay’s Estimate on the Growth of Ideals [8, Theorem 3.3]. Moreover, Gotzmann’s Persistence Theorem [8, Theorem 3.8] asserts that the condition dimk Jr+1 = q(r + 1) suffices to have that the volume polynomial of J is q(t), so the condition we will impose on the points of the grassmannian is dimk Jr+1 6 q(r + 1). We postpone the discussion of the functorial point of view in Section 6, in which we will consider a local ring instead of a field and consequently the general Hilbert scheme Hilbnp(t) n representing the Hilbert functor HilbPp(t) (see [16, Section 4.3]). However in the following A will

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be a finitely generated k-algebra and a local ring with maximal ideal m and residue field K. We will denote by T the polynomial ring A[x0 , . . . , xn ] and for any k-vector space V of dimension N , E will be the tensor product V ⊗k A. ¨ cker coordinates 2. Plu Definition 2.1. (see [16, Section 4.3.3]) Let V be a k-vector space of finite dimension N and p 6 N an integer. Let X be a noetherian scheme over k. The p-Grassmann functor of V is the contravariant functor GrpV (X) : (schemes)◦ → Sets from the category of (schemes) to the category of Sets which associates to X the set of locally free sheaves of rank p, quotient of the free sheaf V ∗ ⊗k OX on X GrpV (X) = {F locally free sheaf of rank p | V ∗ ⊗k OX → F → 0} . This contravariant functor is representable by a scheme called the p-grassmannian of V and that we will denote by Grp (V ). In the case X = Spec A, we have GrpV (Spec A) = {M free A-module of rank p | ∃ F ⊂ E = V ⊗k A s.t. M = E/F } . In the following we will write GrpV (A) instead of GrpV (Spec A) and we will say that a free Amodule F ⊂ E of rank q = N − p belongs to GrpV (A) meaning that E/F is a free A-module of rank p. Let us now describe the grassmannian GrpV (A). Considered E/F ∈ GrpV (A), we have the natural morphism E −→ E/F −→ 0, and its p-th exterior power: φ

∧p E −→ ∧p (E/F ) −→ 0. Chosen a basis {e1 , . . . , eN } of E (as a free A-module) and considered the family of all sets of p indices I = {i1 , . . . , ip }, 1 6 i1 < . . . < ip 6 N , the Pl¨ ucker coordinates of E/F are the elements ∆I = φ(ei1 , . . . , eip ) ∈ E/F ≃ A,

∀ I.

(1)

This set of coordinates gives the usual projective Pl¨ ucker embedding: ϕ : GrpV (A) −→ P (∧p E) E/F 7−→ [. . . : ∆I (E/F ) : . . .]

(2)

Given a basis {λ1 , . . . , λp } of the dual space F ∗ = hλ ∈ E ∗ | λ(F ) = 0i, Pl¨ ucker coordinates in (1) can be computed through the following determinants: λ1 (ei1 ) . . . λ1 (eip ) .. .. .. . ∆I = . . . λp (ei1 ) . . . λp (eip )

(3)

We can also consider another system of coordinates associated to the embedding of the grassmannian GrpV (A) into the projective space P (∧q E): ψ:

GrpV (A) −→ P (∧q E) F = hf1 , . . . , fq i ⊂ E 7−→ f1 ∧ · · · ∧ fq

(4)

where q = N − p and {f1 , . . . , fq } is a basis of F ∈ GrpV (A) as a free A-module of rank q. In this case, the Pl¨ ucker coordinates of F ∈ GrpV (A) can be determined by computing the q × q minors of the q × N matrix obtained putting on the rows the coefficients of the vectors of any base of F in the canonical basis of E. For every multi-index K = {k1 , . . . , kq }, 1 6 k1 < . . . < kq 6 N , we will denote with ΘK ∈ A the Pl¨ ucker coordinate corresponding P to the determinant of the q × q matrix composed by the columns k1 , . . . , kq , so that ψ(F ) = K ΘK (F ) ek1 ∧ · · · ∧ ekq .


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Given any (even not-ordered) set of indices H = {h1 , . . . , hp }, we will denote with ∆H (resp. ΘH ) the determinant of the matrix of the evaluation on the vectors (eh1 , . . . , ehp ) (resp. the determinant of the matrix obtained considering the columns h1 , . . . , hq ). It is easy to check that ∆H = εH ∆H (resp. ΘH = εH ΘH ), where εH is the signature of the permutation σ that orders H, H = σ(H) is the corresponding ordered multi-index and ∆H (resp. ΘH ) is a Pl¨ ucker coordinate. From now on, we will always consider ordered multi-indices. Given two multi-indices K = {k1 , . . . , ka } and H = {h1 , . . . , hb }, we will denote by K|H the set of indices {k1 , . . . , ka , h1 , . . . , hb }, that in general will not be ordered, whereas we will denote with the union K ∪ H the ordered multi-index containing the indices belonging to both K and H. For instance given K = {1, 5} and H = {2}, K|H = {1, 5, 2}, H|K = {2, 1, 5} and K ∪ H = H ∪ K = {1, 2, 5}. Coming back to Pl¨ ucker coordinates the following relation holds: ∆K|H = εK|H ∆K∪H

(resp. ΘK|H = εK|H ΘK∪H ).

Definition 2.2 ([2]). Given a field extension k ⊂ K, an extensor of step l in V (K) = V ⊗k K is an element of ∧l V (K) of the form v1 ∧ · · · ∧ vl with v1 , . . . , vl in V (K). Given a vector subspace F ⊂ V (K) of dimension l, we define an extensor associated to F as an element of the form f1 ∧ · · · ∧ fl ∈ ∧l F , where {f1 , . . . , fl } is a basis of F . Note that all the extensors associated to F are equal up to multiplication by a non-zero scalar. In the following, as done in [2], we will identify a vector subspace F of V (K) of dimension l with an extensor of step l associated to it. We recall Proposition 4.2 of [2]. Proposition 2.3. Given a field extension k ⊂ K and the vectors a1 , . . . , ap , b1 , . . . , bq in V (K) = V ⊗k K with p + q > N , let us consider the extensors T = a1 ∧ · · · ∧ ap and U = b1 ∧ · · · ∧ bq . ×N Let [ ] : V (K) → K be any alternating N -linear form, so that we have the following diagram: ×N V (K)

The following identity holds: P

H={h1 ,...,hN−q } K={k1 ,...,kp+q−N } H∪K={1,...,p}

=

P

[]

K

∧N V (K)

εH|K [ah1 ∧ · · · ∧ ahN−q , U ] ak1 ∧ · · · ∧ akp+q−N =

(5)

εI|J [T, bj1 ∧ · · · ∧ bjN−q ] bi1 ∧ · · · ∧ bip+q−N

(6)

I={i1 ,...,ip+q−N } J ={j1 ,...,jN−q } I∪J ={1,...,q}

We denote these two terms by T ∗ U and we will call ∗ the meet operator associated to [ ]. Proposition 2.4 ([2]). Let k ⊂ K be a field and T and U be two extensors associated to two vector subspaces F ∈ GrpV (K) and G ∈ GrqV (K) (F ⊂ V ⊗k K and G ⊂ V ⊗k K) such that p + q > N (i.e F ∪ G spans the vector space V (K) = V ⊗k K and F ∩ G 6= 0). Let ∗ be the ∗ meet operator associated to any N -linear form [ ] which consists of a basis of ∧N V (K) ≃ K. Then T ∗ U is an extensor associated to the vector subspace F ∩ G of dimension p + q − N . Proof. See [2, Proposition 4.3].

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Proposition 2.5. Let F ∈ GrpV (A) be a free submodule of E, q = N − p and 1 6 m 6 q. Let [. . . : ∆I (F ) : . . .] be its coordinates with respect to the Pl¨ ucker embedding given in (2), then the elements X (7) εH|I ∆I (F ) eh1 ∧ · · · ∧ ehm , ∀ J = {j1 , . . . , jp+m }, δJm (F ) = I={i1 ,...,ip } H={h1 ,...,hm } H∪I=J

generate ∧m F . Proof. Tensoring by the residue field and using Nakayama’s Lemma, we can assume without loss of generality that A = K is a field. Let us use equation (6) with T = F ∈ GrpV (K) and U = BJ = ej1 ∧· · ·∧ejp+m ∈ GrVN −p−m (K). By Proposition 2.4, we have that X εH|I [F, BI ] BH F ∗ BJ = I={i1 ,...,ip } H={h1 ,...,hm } H∪I=J

is an extensor associated to F ∩ hBJ i in V (K). If we take the N -linear form [ ] equal to the determinant in a basis of E which consists of the union of a basis of F and a basis of E/F , then [F, BI ] = ∆I (F ), therefore X F ∗ BJ = εH|I ∆I (F ) BH = δJm (F ) I={i1 ,...,ip } H={h1 ,...,hm } H∪I=J

generates ∧m F ∩ hBJ i . Thus the family

δJm (F ) = F ∗ BJ ,

generates

[ J

GrpV

∀ J = {j1 , . . . , jp+m }

∧m F ∩ hBJ i = ∧m F.

Proposition 2.6. Let F ∈ (A) be a free submodule of E, q = N − p and 1 6 m 6 q. Let [. . . : ΘK (F ) : . . .] be its coordinates with respect to the Pl¨ ucker embedding given in (4), then the family of elements X ∀ J = {j1 , . . . , jq−m }, (8) θJm (F ) = εJ |H ΘJ ∪H (F ) eh1 ∧ · · · ∧ ehm , H={h1 ,...,hm }

generate ∧m F . Proof. As done in the proof of Proposition 2.5, tensoring by the residue field K of A and using Nakayama’s Lemma, we can assume A = K. Let I be equal to the P multi-index such that J ∪I q= {1, . . . , N }. Let us apply the meet operator between T = F = K ΘK (F ) ek1 ∧ · · · ∧ ekq ∈ ∧ V (K) and U = BI = ei1 ∧ · · · ∧ eiN−q+m ∈ GrVq−m (K) according to (5) X F ∗ BI = ΘK (F ) ek1 ∧ · · · ∧ ekq ∗ BI = K

=

X

K K=J ∪H

ΘK (F ) εJ |H eh1 ∧ · · · ∧ ehm [ej1 , . . . , ejq−m , ei1 , . . . , eiN−q+m ].


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If we consider a N -linear form [ ] such that [ej1 , . . . , ejq−m , ei1 , . . . , eiN−q+m ] is equal to 1, we obtain X X F ∗ BI = εJ |H ΘJ ∪H (F ) eh1 ∧ · · · ∧ ehm = θJm(F ). ΘK (F ) εJ |H eh1 ∧ · · · ∧ ehm = K K=J ∪H

H

Hence θJm (F ) is an extensor associated to F ∩ hBI i in V (K) (by Proposition 2.4) and generates ∧m F ∩ hBI i . Thus the family θJm (F ), ∀ J = {j1 , . . . , jq−m } generates [ ∧m F ∩ hBI i = ∧m F. J

Example 2.7. Let us apply these results to the grassmannian Gr2k6 (k). Let {e1 , . . . , e6 } be the fixed basis of the vector space E = V ⊗k k = V = k6 and let us consider the vector subspace F = hf1 = e1 , f2 = e2 , f3 = e3 + 2e4 , f4 = e5 − e6 i. A basis for the quotient space E/F is {e4 , e6 } and a basis of F ∗ is {λ1 = e∗5 + e∗6 , λ2 = 2e∗3 − e∗4 }. Applying the method showed in (3), we obtain that ∆35 (F ) = −2,

∆36 (F ) = −2,

∆45 (F ) = 1,

∆46 (F ) = 1

and the other 11 Pl¨ ucker coordinates vanish. Using the other system of coordinates, the 4 coordinates that do not vanish are Θ1235 (F ) = 1,

Θ1236 (F ) = −1,

Θ1245 (F ) = 2,

Θ1246 (F ) = −2.

Let us now apply Proposition 2.5 and Proposition 2.6 to compute two systems of generators of ∧2 F . • By Proposition 2.5, 2 2 2 2 δ1235 (F ) = ∆35 (F )e1 ∧ e2 = −2e1 ∧ e2 = δ1236 (F ) = −2δ1245 (F ) = −2δ1246 (F ), 2 2 δ1345 (F ) = ∆45 (F )e1 ∧ e3 − ∆35 (F )e1 ∧ e4 = e1 ∧ e3 + 2e1 ∧ e4 = δ1346 (F ), 2 2 δ1345 (F ) = −∆36 (F )e1 ∧ e5 + ∆35 (F )e1 ∧ e6 = 2e1 ∧ e5 − 2e1 ∧ e6 = −2δ1456 (F ), 2 2 δ2345 (F ) = ∆45 (F )e2 ∧ e3 − ∆35 (F )e2 ∧ e4 = e2 ∧ e3 + 2e2 ∧ e4 = δ2346 (F ), 2 2 δ2356 (F ) = −∆36 (F )e2 ∧ e5 + ∆35 (F )e2 ∧ e6 = 2e2 ∧ e5 − 2e2 ∧ e6 = −2δ2456 (F ), 2 δ3456 (F ) = −∆46 (F )e3 ∧ e5 + ∆45 (F )e3 ∧ e6 + ∆36 (F )e4 ∧ e5 − ∆35 (F )e4 ∧ e6 = = −e3 ∧ e5 + e3 ∧ e6 − 2e4 ∧ e5 + 2e4 ∧ e6

• By Proposition 2.6, 2 θ12 (F ) = Θ1235 (F )e3 ∧ e5 + Θ1236 (F )e3 ∧ e6 + Θ1245 (F )e4 ∧ e5 + Θ1246 (F )e4 ∧ e6 = = e3 ∧ e5 − e3 ∧ e6 + 2e4 ∧ e5 − 2e4 ∧ e6 , 2 (F ) θ14 , 2 2 2 θ15 (F ) = Θ1235 (F )e2 ∧ e3 + Θ1245 (F )e2 ∧ e4 = e2 ∧ e3 + 2e2 ∧ e4 = −θ16 (F ),

2 θ13 (F ) = −Θ1235 (F )e2 ∧ e5 − Θ1236 (F )e2 ∧ e6 = −e2 ∧ e5 + e2 ∧ e6 =

2 (F ) θ24 , 2 2 2 θ25 (F ) = −Θ1235 (F )e1 ∧ e3 − Θ1245 (F )e1 ∧ e4 = −e1 ∧ e3 − 2e1 ∧ e4 = −θ26 (F ),

2 θ23 (F ) = Θ1235 (F )e1 ∧ e5 + Θ1236 (F )e1 ∧ e6 = e1 ∧ e5 − e1 ∧ e6 =

2 (F ) θ 2 (F ) θ45 = − 46 2 2 We can see that any generator arising with the first method appears also in the second list (at most multiplied by a scalar), so the two spaces generated by these two families of vectors in ∧2 E are the same and equal to ∧2 F . 2 2 θ35 (F ) = Θ1235 (F )e1 ∧ e2 = e1 ∧ e2 = −θ36 (F ) =

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Remark 2.8. It can be easily proved that the two systems of coordinates [. . . : ∆I : . . .] and [. . . : ΘK : . . .] are equivalent, indeed for every multi-index K of q indices, ΘK = −εK|I ∆I , where K ∪ I = {1, . . . , N } and εK|I is the signature of the permutation σ that orders K|I, that is σ(K|I) = K ∪ I. An analogous relation can be extended to the generators δJm (F ) and {θIm (F )} of ∧m F ; indeed if we consider J = {j1 , . . . , jp+m } and I = {i1 , . . . , iq−m } such that J ∪ I = {1, . . . , N }, δJm (F ) = −εJ |I θIm (F ), where as before, εJ |I is the signature of the permutation that orders J |I. For this reason, from now on we will only use the set of Pl¨ ucker coordinates [. . . : ∆I : . . .], describing the embedding of the Grassmannian GrpV (A) in the projective space P(∧p E). Definition 2.9. In the following we will consider as base vector space V = Sr , the homogeneous polynomials of degree r in S, with its standard monomial basis. Given a Hilbert polynomial p(t) with Gotzmann number equal to r, we are interested in subspaces of Sr (more generally p(r) Tr = Sr ⊗k A) of dimension q(r), that is elements of the grassmannian GrSr (k) (more generally p(r)

p(r)

GrSr (A)). Given an ideal I ⊂ S (resp. I ⊂ T ), we will write that I ∈ GrSr (k) (resp. p(r)

GrSr (A)) meaning that the module Sr /Ir (resp. Tr /Ir ) is free of rank p(r) and that I ∈ n n HilbPp(t) (k) (resp. HilbPp(t) (A)) meaning that the module St /It (resp. Tt /It ) is free of rank p(t), ∀ t > r. For every m, elements of the type δJm become X εH|I ∆I xα(h1 ) ∧ · · · ∧ xα(hm ) , δJm = I={i1 ,...,ip } H={h1 ,...,hm } H∪I=J

where

n+r α : 1, . . . , n

−→

is a bijection from the integers, from 1 up to degree r such that

(

(a0 , . . . , an ) ∈ Nn+1

n+r n ,

n X ai = r i=0

)

to the multi-indices defining monomials of n+r n

xα(1) >DegRevLex xα(2) >DegRevLex . . . >DegRevLex xα((

)) .

Moreover we define • •

•

Bm = δJm , ∀ J = {j1 , . . . , jp(r)+m } , X xi δJm = εH|I ∆I xi xα(h1 ) ∧ · · · ∧ xi xα(hm ) , xi B m =

I={i1 ,...,ip } H={h1 ,...,hm } H∪I=J xi δJm , ∀ J

= {j1 , . . . , jp(r)+m } .

3. Iarrobino-Kleiman equations

Theorem 3.1 (Iarrobino, Kleiman [12]). Let p(t) be an admissible Hilbert polynomial in Pn n with Gotzmann number r. HilbPp(t) (k) can be defined as a closed subscheme of the grassmannian p(r)

GrSr (k) by equations of degree q(r + 1) + 1.


9

Proof. Let us fix the usual monomial basis on Sr and the associated Pl¨ ucker coordinates for p(r) GrSr (k) in accord with Definition 2.9 and let q(t) = n+t − p(t). Moreover let I be the ideal n 1 generated in degree r by the polynomials representing the elements δJ : I = (B1 ). n

By Proposition 2.5, we know that dim Ir = q(r), so to obtain equations for HilbPp(t) (k) it suffices to impose dim Ir+1 6 q(r + 1). Let us consider the matrix with n+r+1 columns corresponding n to the monomial basis of Sr+1 and |B1 | · (n + 1) rows corresponding to the polynomials contained in xi B1 , ∀ i = 0, . . . , n. The rows of this matrix span x0 Ir + . . . + xn Ir = Ir+1 , so we obtain the n equation of HilbPp(t) (k) by imposing that all the minors of dimension q(r + 1) + 1 vanish. 2

Example 3.2. Let us now see how to compute the Iarrobino-Kleiman equations for HilbP2 (K) with a field extension k ⊂ K. Since the Gotzmann number is 2, we have to consider the grassmannian Gr2k[x,y,z]2 (K), with k[x, y, z]2 ≃ k6 (as in Example 2.7) with monomial basis {e1 = x2 , e2 = xy, e3 = y 2 , e4 = xz, e5 = yz, e6 = z 2 }. Following the proof of Theorem 3.1, we consider the ideal I = (B1 ) and we impose that any subset of q(3) + 1 = 9 polynomials of {xB1 , yB1 , zB1 } is dependent. For instance, considering the polynomials represented in the following matrix 1 x δ126 1 x δ156 1 x δ234 1 x δ356 1 y δ123 1 y δ345 1 z δ146 1 z δ234 1 z δ456

x3 ∆26 ∆56 0 0 0 0 0 0 0

x2 y −∆16 0 ∆34 0 ∆23 0 0 0 0

xy 2 0 0 −∆24 ∆56 −∆13 0 0 0 0

y3 0 0 0 0 ∆12 ∆45 0 0 0

x2 z 0 0 ∆23 0 0 0 ∆46 0 0

xyz 0 −∆16 0 −∆36 0 −∆35 0 ∆34 0

y2z 0 0 0 0 0 ∆34 0 −∆24 0

xz 2 ∆12 ∆15 0 ∆35 0 0 −∆16 ∆23 ∆56

yz 2 0 0 0 0 0 0 0 0 −∆46

z3 0 0 0 0 0 0 ∆14 0 ∆45

the dependency condition corresponds to the vanishing of the minors of order 9: • − ∆16 ∆23 ∆224 ∆26 ∆35 ∆45 ∆246 + ∆13 ∆16 ∆24 ∆26 ∆34 ∆35 ∆45 ∆246 + ∆15 ∆23 ∆224 ∆26 ∆36 ∆45 ∆246 + − ∆13 ∆15 ∆24 ∆26 ∆34 ∆36 ∆45 ∆246 − ∆216 ∆223 ∆24 ∆26 ∆45 ∆46 ∆56 + ∆13 ∆216 ∆23 ∆24 ∆36 ∆45 ∆46 ∆56 + − ∆12 ∆16 ∆23 ∆26 ∆234 ∆246 ∆56 − ∆12 ∆15 ∆26 ∆334 ∆246 ∆56 + ∆12 ∆15 ∆24 ∆26 ∆34 ∆35 ∆246 ∆56 + + ∆12 ∆16 ∆24 ∆234 ∆35 ∆246 ∆56 − ∆12 ∆16 ∆224 ∆235 ∆246 ∆56 + ∆12 ∆16 ∆23 ∆24 ∆34 ∆36 ∆246 ∆56 + − ∆12 ∆23 ∆224 ∆36 ∆45 ∆246 ∆56 + ∆12 ∆13 ∆24 ∆34 ∆36 ∆45 ∆246 ∆56 + ∆12 ∆216 ∆23 ∆234 ∆46 ∆256 + − ∆12 ∆216 ∆23 ∆24 ∆35 ∆46 ∆256 + ∆212 ∆334 ∆246 ∆256 − ∆212 ∆24 ∆34 ∆35 ∆246 ∆256 , • ∆16 ∆23 ∆224 ∆26 ∆35 ∆245 ∆46 − ∆13 ∆16 ∆24 ∆26 ∆34 ∆35 ∆245 ∆46 − ∆15 ∆23 ∆224 ∆26 ∆36 ∆245 ∆46 + + ∆13 ∆15 ∆24 ∆26 ∆34 ∆36 ∆245 ∆46 + ∆216 ∆223 ∆24 ∆26 ∆245 ∆56 − ∆13 ∆216 ∆23 ∆24 ∆36 ∆245 ∆56 + + ∆12 ∆16 ∆23 ∆26 ∆234 ∆45 ∆46 ∆56 + ∆12 ∆15 ∆26 ∆334 ∆45 ∆46 ∆56 − ∆12 ∆15 ∆24 ∆26 ∆34 ∆35 ∆45 ∆46 ∆56 + − ∆12 ∆16 ∆24 ∆234 ∆35 ∆45 ∆46 ∆56 + ∆12 ∆16 ∆224 ∆235 ∆45 ∆46 ∆56 − ∆12 ∆16 ∆23 ∆24 ∆34 ∆36 ∆45 ∆46 ∆56 + + ∆12 ∆23 ∆224 ∆36 ∆245 ∆46 ∆56 − ∆12 ∆13 ∆24 ∆34 ∆36 ∆245 ∆46 ∆56 + ∆14 ∆16 ∆223 ∆24 ∆26 ∆45 ∆256 + − ∆12 ∆216 ∆23 ∆234 ∆45 ∆256 + ∆12 ∆216 ∆23 ∆24 ∆35 ∆45 ∆256 − ∆13 ∆14 ∆16 ∆23 ∆24 ∆36 ∆45 ∆256 + − ∆212 ∆334 ∆45 ∆46 ∆256 + ∆212 ∆24 ∆34 ∆35 ∆45 ∆46 ∆256 − ∆12 ∆14 ∆16 ∆23 ∆234 ∆356 + + ∆12 ∆14 ∆16 ∆23 ∆24 ∆35 ∆356 , • − ∆14 ∆16 ∆223 ∆24 ∆26 ∆45 ∆46 ∆56 + ∆13 ∆14 ∆16 ∆23 ∆24 ∆36 ∆45 ∆46 ∆56 + ∆12 ∆14 ∆16 ∆23 ∆234 ∆46 ∆256 + − ∆12 ∆14 ∆16 ∆23 ∆24 ∆35 ∆46 ∆256 , • ∆14 ∆16 ∆323 ∆26 ∆45 ∆46 ∆56 + ∆14 ∆15 ∆223 ∆26 ∆34 ∆45 ∆46 ∆56 − ∆13 ∆14 ∆16 ∆23 ∆34 ∆35 ∆45 ∆46 ∆56 + − ∆13 ∆14 ∆16 ∆223 ∆36 ∆45 ∆46 ∆56 + ∆12 ∆14 ∆16 ∆223 ∆35 ∆46 ∆256 − ∆12 ∆14 ∆223 ∆34 ∆45 ∆46 ∆256 ,

(9)

10


• −

∆14 ∆15 ∆223 ∆24 ∆26 ∆45 ∆46 ∆56

+ ∆13 ∆14 ∆16 ∆23 ∆24 ∆35 ∆45 ∆46 ∆56 − ∆12 ∆14 ∆16 ∆223 ∆34 ∆46 ∆256 +

+ ∆12 ∆14 ∆223 ∆24 ∆45 ∆46 ∆256 , • − ∆14 ∆16 ∆23 ∆224 ∆26 ∆35 ∆45 ∆46 + ∆13 ∆14 ∆16 ∆24 ∆26 ∆34 ∆35 ∆45 ∆46 + ∆14 ∆15 ∆23 ∆224 ∆26 ∆36 ∆45 ∆46 + − ∆13 ∆14 ∆15 ∆24 ∆26 ∆34 ∆36 ∆45 ∆46 − ∆12 ∆14 ∆16 ∆23 ∆26 ∆234 ∆46 ∆56 − ∆12 ∆14 ∆15 ∆26 ∆334 ∆46 ∆56 + + ∆12 ∆14 ∆15 ∆24 ∆26 ∆34 ∆35 ∆46 ∆56 + ∆12 ∆14 ∆16 ∆24 ∆234 ∆35 ∆46 ∆56 − ∆12 ∆14 ∆16 ∆224 ∆235 ∆46 ∆56 + + ∆12 ∆14 ∆16 ∆23 ∆24 ∆34 ∆36 ∆46 ∆56 − ∆12 ∆14 ∆23 ∆224 ∆36 ∆45 ∆46 ∆56 + ∆12 ∆13 ∆14 ∆24 ∆34 ∆36 ∆45 ∆46 ∆56 + + ∆212 ∆14 ∆334 ∆46 ∆256 − ∆212 ∆14 ∆24 ∆34 ∆35 ∆46 ∆256 , • ∆14 ∆16 ∆323 ∆26 ∆34 ∆46 ∆56 + ∆14 ∆15 ∆223 ∆26 ∆234 ∆46 ∆56 − ∆14 ∆15 ∆223 ∆24 ∆26 ∆35 ∆46 ∆56 + − ∆13 ∆14 ∆16 ∆23 ∆234 ∆35 ∆46 ∆56 + ∆13 ∆14 ∆16 ∆23 ∆24 ∆235 ∆46 ∆56 − ∆13 ∆14 ∆16 ∆223 ∆34 ∆36 ∆46 ∆56 + − ∆12 ∆14 ∆223 ∆234 ∆46 ∆256 + ∆12 ∆14 ∆223 ∆24 ∆35 ∆46 ∆256 , • − ∆14 ∆16 ∆223 ∆24 ∆26 ∆35 ∆45 ∆46 + ∆14 ∆15 ∆223 ∆24 ∆26 ∆36 ∆45 ∆46 + ∆12 ∆14 ∆16 ∆23 ∆234 ∆35 ∆46 ∆56 + − ∆12 ∆14 ∆16 ∆23 ∆24 ∆235 ∆46 ∆56 + ∆12 ∆14 ∆16 ∆223 ∆34 ∆36 ∆46 ∆56 − ∆12 ∆14 ∆223 ∆24 ∆36 ∆45 ∆46 ∆56 , • ∆13 ∆14 ∆16 ∆23 ∆24 ∆26 ∆35 ∆45 ∆46 − ∆13 ∆14 ∆15 ∆23 ∆24 ∆26 ∆36 ∆45 ∆46 − ∆12 ∆14 ∆16 ∆223 ∆26 ∆34 ∆46 ∆56 + − ∆12 ∆14 ∆15 ∆23 ∆26 ∆234 ∆46 ∆56 + ∆12 ∆14 ∆15 ∆23 ∆24 ∆26 ∆35 ∆46 ∆56 + ∆12 ∆13 ∆14 ∆23 ∆24 ∆36 ∆45 ∆46 ∆56 + + ∆212 ∆14 ∆23 ∆234 ∆46 ∆256 − ∆212 ∆14 ∆23 ∆24 ∆35 ∆46 ∆256 , • − ∆13 ∆14 ∆216 ∆23 ∆24 ∆35 ∆45 ∆46 + ∆13 ∆14 ∆15 ∆16 ∆23 ∆24 ∆36 ∆45 ∆46 + ∆12 ∆14 ∆216 ∆223 ∆34 ∆46 ∆56 + + ∆12 ∆14 ∆15 ∆16 ∆23 ∆234 ∆46 ∆56 − ∆12 ∆14 ∆15 ∆16 ∆23 ∆24 ∆35 ∆46 ∆56 − ∆12 ∆14 ∆16 ∆223 ∆24 ∆45 ∆46 ∆56 .

4. Bayer equations To reduce the degree of equations and to reach the bound given by Bayer in [3], we collect together rows in the matrix of polynomials in degree r + 1 that come from polynomials in degree r multiplied by the same variable and to express the minors of such submatrices as linear combinations of Pl¨ ucker coordinates. For instance in Example 3.2, we may express the minors of the matrix in (9), putting together the first 4 rows (representing 4 generators multiplied by x), the fifth and the sixth (corresponding to 2 generators multiplied by y) and the last 3 (3 generators multiplied by z). In this way we obtain equations of degree 3. Theorem 4.1 (Bayer [3]). Let p(t) be an admissible Hilbert polynomial in Pn with Gotzmann n p(r) number r. HilbPp(t) (k) can be defined as a closed subscheme of the grassmannian GrSr (k) by equations of degree n + 1. p(r) Proof. Let us consider the point I = hB1 i of GrSr (k) and let q(t) = n+t n − p(t). The condition n on the dimension of Ir+1 , that we impose to obtain equations of HilbPp(t) (k), can be rewritten as ⇐⇒

dimk Ir+1 6 q(r + 1) In Theorem 3.1, we look at ∧q(r+1)+1 Ir+1 =

∧q(r+1)+1 I *q(r+1)+1 ^

r+1

∧q(r+1)+1 Ir+1 = 0.

(10)

as

li δJ1 i

i=1

But to compute ∧q(r+1)+1 Ir+1 , we can take indeed q(r+1)+1 ∧ Ir+1 = x0 δJm00 ∧ · · · ∧ xn δJmnn

∀ li ∈ {x0 , . . . , xn } ∀ multi-index Ji of size p(r) + 1

+

.

advantage of the generators of ∧m Ir+1 for every m,

∀ x0 δJm00 ∈ x0 Bm0 P , . . . , xn δJmnn ∈ xn Bmn ∀ mi 6 q(r) s.t. i mi = q(r + 1) + 1

.

Every generator of this type has coefficients represented by homogeneous polynomials in the Pl¨ ucker coordinates of degree n + 1. By imposing that all these coefficients vanish, we determine n p(r) equations of degree n + 1 for HilbPp(t) (k) as a closed subscheme of GrSr (k).


11 2

Example 4.2. Let us examine again the case of the Hilbert scheme HilbP2 (K) with a field extension k ⊂ K and the same notation defined in Example 3.2. Since q(2) = 4 and q(3) + 1 = 9, we have to compute the exterior products x δJm00 ∧ y δJm11 ∧ z δJm22 for (m0 , m1 , m2 ) = (4, 4, 1), (4, 1, 4), (1, 4, 4), (4, 3, 2), (4, 2, 3), (3, 4, 2), (3, 2, 4), (2, 4, 3), (2, 3, 4), (3, 3, 3). For instance 4 2 3 let us write explicitly the value of x δ123456 ∧ y δ1346 ∧ z δ23456 : 4 x δ123456 = ∆56 x3 ∧ x2 y ∧ xy 2 ∧ x2 z − ∆46 x3 ∧ x2 y ∧ xy 2 ∧ xyz + ∆45 x3 ∧ x2 y ∧ xy 2 ∧ xz 2 +

+ ∆36 x3 ∧ x2 y ∧ x2 z ∧ xyz − ∆35 x3 ∧ x2 y ∧ x2 z ∧ xz 2 + ∆34 x3 ∧ x2 y ∧ xyz ∧ xz 2 + − ∆26 x3 ∧ xy 2 ∧ x2 z ∧ xyz + ∆25 x3 ∧ xy 2 ∧ x2 z ∧ xz 2 − ∆24 x3 ∧ xy 2 ∧ xyz ∧ xz 2 + + ∆23 x3 ∧ x2 z ∧ xyz ∧ xz 2 + ∆16 x2 y ∧ xy 2 ∧ x2 z ∧ xyz − ∆15 x2 y ∧ xy 2 ∧ x2 z ∧ xz 2 + + ∆14 x2 y ∧ xy 2 ∧ xyz ∧ xz 2 − ∆13 x2 y ∧ x2 z ∧ xyz ∧ xz 2 + ∆12 xy 2 ∧ x2 z ∧ xyz ∧ xz 2 , 2 y δ1346 = ∆46 x2 y ∧ y 3 − ∆36 x2 y ∧ xyz + ∆34 x2 y ∧ yz 2 + ∆16 y 3 ∧ xyz − ∆14 y 3 ∧ yz 2 + ∆13 xyz ∧ yz 2 , 3 z δ23456 = ∆56 xyz ∧ y 2 z ∧ xz 2 − ∆46 xyz ∧ y 2 z ∧ yz 2 + ∆45 xyz ∧ y 2 z ∧ z 3 + ∆36 xyz ∧ xz 2 ∧ yz 2 +

− ∆35 xyz ∧ xz 2 ∧ z 3 + ∆34 xyz ∧ yz 2 ∧ z 3 − ∆26 y 2 z ∧ xz 2 ∧ yz 2 + ∆25 y 2 z ∧ xz 2 ∧ z 3 + − ∆24 y 2 z ∧ yz 2 ∧ z 3 + ∆23 xz 2 ∧ yz 2 ∧ z 3 .

The coefficients of the exterior product of these three elements are polynomials in the Pl¨ ucker 2 coordinates of degree 3 that belong to the ideal defining the Hilbert scheme HilbP2 (K): • − ∆226 ∆46 + ∆25 ∆246 + ∆16 ∆26 ∆56 − ∆14 ∆256 , • + ∆25 ∆26 ∆46 − ∆25 ∆45 ∆46 − ∆16 ∆25 ∆56 , • − ∆24 ∆26 ∆46 + ∆16 ∆24 ∆56 + ∆14 ∆45 ∆56 , • + ∆23 ∆26 ∆46 + ∆25 ∆34 ∆46 − ∆16 ∆23 ∆56 − ∆14 ∆35 ∆56 , • − ∆24 ∆25 ∆46 + ∆14 ∆25 ∆56 , • + ∆16 ∆24 ∆45 + ∆14 ∆245 + ∆224 ∆46 − ∆14 ∆25 ∆46 , • + ∆25 ∆26 ∆34 − ∆24 ∆25 ∆36 − ∆25 ∆34 ∆45 + ∆13 ∆25 ∆56 , • + ∆16 ∆24 ∆35 − ∆14 ∆25 ∆36 + ∆14 ∆35 ∆45 + ∆23 ∆24 ∆46 , • − ∆16 ∆24 ∆25 + ∆14 ∆25 ∆26 − ∆14 ∆25 ∆45 , • + ∆15 ∆16 ∆24 − ∆14 ∆16 ∆25 + ∆14 ∆15 ∆45 − ∆12 ∆24 ∆46 .

5. BLMR equations Throughout last two sections, given an admissible Hilbert polynomial p(t) on Pn with Gotz− p(t), we mann number r and degree d and the associated volume polynomial q(t) = n+t n set n−d−1+t n+t n−d−1+t ′ q (t) = q(t) − = − − p(t), (11) n−d−1 n n−d−1 so that q(t) − q ′ (t) = dimk k[xd+1 , . . . , xn ]t . p(r)

Proposition 5.1. Let U ′ be the set of all the elements Ir ∈ GrSr (k) (i.e. Sr /Ir has dimension p(r)) such that Ir has a set of generators of the type: GrI = xα + fα | xα ∈ k[xd+1 , . . . , xn ]r and fα ∈ (xd , . . . , x0 ) ∪ gj | gj ∈ (xd , . . . , x0 ) (12)

where the first set of generators contains by construction q(r) − q ′ (r) elements and the second set has q ′ (r) polynomials, so that dimk Ir = q(r). p(r) that Then U ′ is a non-empty open subset in GrSr (k) and Ir+1 has a set of generators Gr+1 I can be represented by a matrix of the type:   • • • Id  0 D1 (13) Ar+1 =  0 D2 where:

12


• the columns belonging to the left part of the matrix correspond to the monomials in k[xd+1 , . . . , xn ]r+1 and the columns on the right to the monomials in (x0 , . . . , xd )r+1 ; • the top-left submatrix Id is the identity matrix of order q(r + 1) − q ′ (r + 1); • the rows of D1 contain the coefficients of all the generators multiplied by a variable xh , h = 0, . . . , d; • the rows of D2 contain the coefficients of the generators gj multiplied by a variable xh , h = ′ d + 1, . . . , n and the coefficients of the polynomials xi′ fα′ − xi fα such that xi′ xα = xi xα and i, i′ > d + 1. Moreover the subset U ⊂ U ′ of all the ideals Ir such that rank D1 > q ′ (r + 1) is open and p(r) U pgl = {gU | g ∈ pgl(n + 1)} is an open covering of GrSr (k). Proof. Let us consider the canonical projection π : k[x0 , . . . , xn ]r −→ (k[x0 , . . . , xn ]/(x0 , . . . xd ))r ≃ k[xd+1 , . . . , xn ]r . p(r)

p(r)

The subset U ′ of GrSr (k) is open because U ′ = π −1 (k[xd+1 , . . . , xn ]r ) ∩ GrSr (k). Moreover U ′ p(r)

is non-empty because any Borel ideal J defining a point of GrSr (k) (i.e. dimk Jr = q(r)) belongs to U ′ . Indeed, dimk Jt > q(t), ∀ t > r (by Macaulay’s Estimate on the Growth of Ideals) implies that the Hilbert polynomial of k[x0 , . . . , xn ]/J has degree smaller than or equal to deg p(t) = d and so k[xd+1 , . . . , xn ]>r ⊆ J (see [5, Proposition 2.3]). Therefore every ideal Ir ∈ U ′ has a basis GrI (of Ir as k-vector space) as the one described in (12). S we are looking A set of generators for Ir+1 is S1 · GrI = i {xi GrI }. The set of generators Gr+1 I for can be easily obtained from S1 · GrI just modifying few elements: for every monomial xγ in k[xd+1 , . . . , xn ]r+1 we choose only one product xi (xα + fα ) such that xγ = xi xα , to be left (and corresponding to a row in the first block of Ar+1 ), whereas we replace any other in Gr+1 I ′ ′ polynomial xi′ (xα + fα′ ), such that xγ = xi′ xα , by xi′ fα′ − xi fα (which belongs to (x0 , . . . , xd ) and corresponds to a row of D2 ). Obviously the condition rank D1 > q ′ (r + 1) is an open condition and we call U ⊂ U ′ the corresponding open subset. Again this open subset is not empty because it contains for instance p(r) all the subspaces Jr defined by a Borel ideal J ∈ GrSr (k). To prove the last statement, we consider any term ordering such that xn ≻ . . . ≻ x0 , and we recall that in general coordinates the initial ideal of any ideal is Borel-fixed (see [6, Theorem 15.20]). Then for a general g ∈ pgl(n + 1), J = (in(gI)r ) is Borel. Note that J belongs to n p(r) / HilbPp(t) (k), J can differ from in(gI). As J is Borel-fixed GrSr (k), but if I ∈ dimk (x0 Jr + . . . + xd Jr ) = dimk Jr+1 ∩ (x0 , . . . , xd ) = = dimk Jr+1 − dimk k[xd+1 , . . . , xn ]r+1 > q ′ (r + 1), hence

dimk x0 (gI)r + . . . + xd (gI)r > dimk x0 in(gI)r + . . . + xd in(gI)r = = dimk (x0 Jr + . . . + xd Jr ) > q ′ (r + 1).

Finally we can conclude that gI ∈ U because a set of generators of the vector space x0 (gI)r + . . . + xd (gI)r corresponds to the rows of D1 . D1 Making reference to the above matrix Ar+1 , let us denote by D the submatrix . D2 Corollary 5.2. Let I ⊂ S be any ideal belonging to U . Then n (1) I ∈ U ∩ HilbPp(t) (k) ⇐⇒ rank D = rank D1 = q ′ (r + 1); n (2) I ∈ / U ∩ HilbPp(t) (k) ⇐⇒ either rank D1 > q ′ (r + 1) or rank D1 = q ′ (r + 1) < rank D.


13

n

Proof. (1) follows from the fact that I ∈ HilbPp(t) (k) if and only if dimk Ir+1 = rank Ar+1 = q(r + 1) and from the special form of Ar+1 . (2) is another way to write (1). Theorem 5.3. Let p(t) be an admissible Hilbert polynomial in Pn of degree d and Gotzmann n p(r) number r. HilbPp(t) (k) can be defined as a closed subscheme of the grassmannian GrSr (k) by equations of degree smaller than or equal to d + 2. Proof. We divide the proof in two steps. Step 1. First of all we construct a set of polynomials T in k[∆] such that given an ideal I ∈ U , every polynomial in T vanishes on the Pl¨ ucker coordinates [. . . : ∆(I) : . . .] of I if and only if n I ∈ U ∩ HilbPp(t) (k). Let us choose in all the possible ways a set of d + 1 elements of the type xi δJmii such that i = P 0, . . . , d and i mi = q ′ (r + 1). Moreover let us consider any multi-index K = {k1 , . . . , kp(r)+1 }, such that the corresponding monomials {xα(k1 ) , . . . , xα(kp(r)+1 ) } of degree r do not belong to k[xd+1 , . . . , xn ]r , and any variable xj , j = d + 1, . . . , n. A first part T ′ of the polynomials in T is represented by all the coefficients of all exterior products of the type   ^ 1  xi δJmii  ∧ xj δK . (14) i=0,...,d

Afterwards let us choose in all the possible ways a multi-index H = {h1 , . . . , hp(r) }, such that the corresponding monomials {xα(h1 ) , . . . , xα(hp(r) ) } of degree r do not belong to k[xd+1 , . . . , xn ]r ′ and a monomial xγ ∈ k[xd+1 , . . . , xn ]r−1 with two indices j, j ∈ {d + 1, . . . , n}. The second part T ′′ of the polynomials in T can be obtained collecting the coefficients of all the exterior products of the type   ^ 1 1  ) (15) − xj δH xi δJmii  ∧ (xj ′ δH 2 1 i=0,...,d

′

′

where, called h and h the indices such that xα(h) = xj xγ and xα(h ) = xj ′ xγ , H1 = H ∪ {h} = ′

′

{h, h1 , . . . , hp(r) } and H2 = H ∪ {h } = {h , h1 , . . . , hp(r) }. Note that the conditions given by the vanishing of the polynomials in T = T ′ ∪ T ′′ mean that q ′ (r + 1) rows in the matrix D1 and one rows in the matrix D2 are linearly dependent (directly by the construction of the matrix Ar+1 in Proposition 5.1). These conditions ensure also that q ′ (r + 1) + 1 rows of the matrix D1 are dependent, because of the well-known property of vector spaces saying that s + 1 vectors v1 , . . . , vs+1 , such that every subset of s elements is linear dependent with any other vector u 6= 0, are dependent. The only delicate issue, that we will discuss later in Remark 5.4, is checking that D2 is not a zero matrix. n By Corollary 5.2, I ∈ U belongs to HilbPp(t) (k) if and only if the polynomials of T vanish on [. . . : ∆(I) : . . .]. Note that the coefficients of the exterior products in (14) and (15) are polynomials of degree 6 d + 2 (more precisely the degree is the number of non-zero mi ). p(r)

ucker Step 2. Let I be an element of GrSr (k) and g = (gi,j ) be an element of pgl(n+1). The Pl¨ p(r)

coordinates [. . . : ∆(gI) : . . .] of gI ∈ GrSr (k) are bi-homogeneous polynomials of degree 1 in the Pl¨ ucker coordinates [. . . : ∆(I) : . . .] and of degree q(r) · r in the coefficients gi,j of the matrix g. So given a homogeneous polynomial P of degree s 6 d + 2 in T , P ([. . . : ∆(gI) : . . .]) is a bi-homogeneous polynomial of degree s in [. . . : ∆(I) : . . .] and of degree q(r) · r · s in gi,j . At this point we collect, and denote by CP , the homogeneous polynomials of degree s 6 d + 2 in the Pl¨ ucker coordinates [. . . : ∆(I) : . . .], that spring up as coefficients of P ([. . . : ∆(gI) : . . .]), viewed as a homogeneous polynomial of degree q(r) · r · s in the variables gi,j .

14

J. BRACHAT, P. LELLA, B. MOURRAIN, AND M. ROGGERO n

From Proposition 5.1 and Corollary 5.2, I belongs to HilbPp(t) (k) if and only if for a generic changes of variables g ∈ pgl(n + 1) n

gI ∈ U ∩ HilbPp(t) (k) i.e. all the homogeneous polynomials P ∈ T vanish at [. . . : ∆(gI) : . . .], or equivalently all the coefficients CP for P ∈ T vanish at [. . . : ∆(I) : . . .]. n

p(r)

We finally proved that I ∈ GrSr (k) belongs to HilbPp(t) (k) if and only if [. . . : ∆(I) : . . .] satisfies all the equations of the set [ CP (16) P ∈T

which consists of homogeneous polynomial of degree smaller than or equal to d + 2.

Remark 5.4. Note that a necessary condition in order that D2 is empty is that Ir has no generators belonging to GrI of the type gj and now we prove that it is not possible. For the sake of simplicity, we can think about the monomial ideal obtained in the case fα = gj = 0, ∀ α, ∀ j. Such an ideal should have a Hilbert polynomial pe(t) such that pe(r) = n+r − n−d+r n n−d . Let us show that pe(r) can not be equal to P (r) with P a Hilbert polynomial with Gotzmann number equal to r and of degree d. The first point is to compute the maximal value in degree r of a Hilbert polynomial of degree d and Gotzmann number r. By the decomposition of Hilbert polynomials given by Gotzmann’s Regularity Theorem [8, Theorem 3.11], we know that among the Hilbert polynomials of degree d and Gotzmann number r there is t+d t+d−1 t + d − (r − 1) p(t) = + + ... + , d d d and that any other Hilbert polynomial has at least one binomial coefficient t+(d−i)−j replacing d−i r+(d−i)−j t+d−j r+d−j (i, j > 0). Because of > , the maximal value reached is d d d−i r X d+i P = max {P (r) | P (t) of degree d and Gotzmann number r} = . d i=1

Finally, starting from the decomposition [ k[x0 , . . . , xn ]r = k[x0 , . . . , xd ]i · k[xd+1 , . . . , xn ]r−i , i=0,...,r

we have that X X r r n+r n−d+r+1 d+i n−d+1+r−i d+i pe(r) = − = > = P. n n−d+1 d n−d+1 d i=1

i=1

Example 5.5. Let us apply Theorem 5.3 to the case already considered in Example 3.2 and Example 4.2. Since the Hilbert polynomial is constant, to compute the first set of equations T ′ 4 1 , yδ 1 not (14), we have to consider the wedge product between zδ123456 and the 2 elements xδ456 456 containing monomials in k[x, y]. We obtain 12 polynomials: • • • • • •

− ∆26 ∆46 + ∆45 ∆46 + ∆16 ∆56 , − ∆24 ∆46 + ∆14 ∆56 , − ∆24 ∆45 + ∆12 ∆56 , − ∆36 ∆46 + ∆26 ∆56 + ∆45 ∆56 , − ∆245 − ∆34 ∆46 + ∆24 ∆56 , − ∆25 ∆45 + ∆23 ∆46 ,

• • • • • •

− ∆245 + ∆25 ∆46 − ∆15 ∆56 , ∆34 ∆45 + ∆23 ∆46 − ∆13 ∆56 , ∆14 ∆45 − ∆12 ∆46 , ∆35 ∆46 − ∆25 ∆56 , ∆35 ∆45 − ∆23 ∆56 , ∆15 ∆45 − ∆13 ∆46 + ∆12 ∆56 .


15

To compute the second set of equations T ′′ (15), we have to consider the coefficients of the wedge 4 product between zδ123456 and one element belonging to the set 1 1 1 1 1 1 1 1 1 1 1 1 {yδ145 − xδ245 , yδ245 − xδ345 , yδ146 − xδ246 , yδ246 − xδ346 , yδ156 − xδ256 , yδ256 − xδ356 }. We obtain other 36 generators:

• • • • • • • • • • • • • • • • • •

∆16 ∆25 − ∆15 ∆26 − ∆24 ∆26 + ∆14 ∆36 , − ∆15 ∆24 − ∆224 + ∆14 ∆25 + ∆14 ∆34 , − ∆14 ∆23 + ∆12 ∆25 , − ∆25 ∆26 − ∆26 ∆34 + ∆16 ∆35 + ∆24 ∆36 , − ∆24 ∆25 + ∆14 ∆35 , − ∆23 ∆24 + ∆12 ∆35 , − ∆24 ∆46 + ∆14 ∆56 , − ∆16 ∆24 + ∆14 ∆26 − ∆14 ∆45 , ∆224 − ∆14 ∆25 + ∆12 ∆26 , − ∆226 + ∆16 ∆36 − ∆34 ∆46 + ∆24 ∆56 , − ∆24 ∆26 + ∆14 ∆36 − ∆24 ∆45 , − ∆24 ∆25 + ∆24 ∆34 + ∆12 ∆36 , ∆226 − ∆16 ∆36 − ∆25 ∆46 + ∆15 ∆56 , ∆24 ∆26 − ∆16 ∆34 − ∆15 ∆45 , ∆16 ∆23 − ∆15 ∆25 + ∆24 ∆25 , − ∆35 ∆46 + ∆25 ∆56 , − ∆26 ∆34 + ∆24 ∆36 − ∆25 ∆45 , − ∆225 + ∆23 ∆26 + ∆24 ∆35 ,

• • • • • • • • • • • • • • • • • •

∆24 ∆25 − ∆14 ∆35 , ∆15 ∆23 + ∆23 ∆24 − ∆13 ∆25 , ∆13 ∆14 − ∆12 ∆15 − ∆12 ∆24 , ∆225 + ∆25 ∆34 − ∆15 ∆35 − ∆24 ∆35 , ∆23 ∆25 + ∆23 ∆34 − ∆13 ∆35 , ∆13 ∆24 − ∆12 ∆25 − ∆12 ∆34 , ∆16 ∆25 − ∆15 ∆26 + ∆24 ∆45 , ∆16 ∆23 − ∆13 ∆26 − ∆24 ∆34 + ∆14 ∆35 , ∆14 ∆15 − ∆12 ∆16 − ∆14 ∆24 , ∆25 ∆26 − ∆15 ∆36 + ∆34 ∆45 , ∆23 ∆26 − ∆234 + ∆24 ∆35 − ∆13 ∆36 , ∆15 ∆24 − ∆12 ∆26 − ∆14 ∆34 , − ∆25 ∆26 + ∆16 ∆35 + ∆25 ∆45 , − ∆23 ∆26 − ∆25 ∆34 + ∆15 ∆35 , ∆215 − ∆13 ∆16 − ∆14 ∆25 + ∆12 ∆26 , ∆26 ∆35 − ∆25 ∆36 + ∆35 ∆45 , ∆25 ∆35 − ∆34 ∆35 − ∆23 ∆36 , ∆15 ∆25 − ∆13 ∆26 − ∆14 ∆35 + ∆12 ∆36 .

Now we need to introduce the action of pgl(3) on k[x, y, z] and to understand how the induced action on Gr4k[x,y,z]2 (K) works. Given an element g = (gi,j ) ∈ pgl(3) and its action      x g11 g12 g13 x  y  ←−  g21 g22 g23   y  , z g31 g32 g33 z

the induced action on the k-vector space k[x, y, z]2 is represented by the matrix        

x2 xy y2 xz yz z2

 2 g11  g11 g21    2    ←−  g21  g11 g31     g21 g31  2 g31 

2g11 g12 g12 g21 + g11 g22 2g21 g22 g12 g31 + g11 g32 g22 g31 + g21 g32 2g31 g32

2 g12 g12 g22 2 g22 g12 g32 g22 g32 2 g32



2 g13 g13 g23 2 g23 g13 g33 g23 g33 2 g33

       

x2 xy y2 xz yz z2



   .   

To write explicitly the action of g on the Pl¨ ucker coordinates, we can consider the element 4 , substitute each element xβ of the basis of k[x, y, z]2 with g.xβ obtaining the exterior δ123456 4 product g.δ123456 and then the action is determined by looking at the coefficients of the same 4 4 . For instance the Pl¨ ucker coordinate and g.δ123456 element of the basis of ∧4 k[x, y, z]2 in δ123456 2 2 2 4 ∆25 , coefficient of x ∧ y ∧ xz ∧ z in δ123456 , will be send by the action of g to the coefficient of 4 . x2 ∧ y 2 ∧ xz ∧ z 2 in g.δ123456 Finally, for every polynomial P contained in T , we have to compute the action of g, that is substituting each ∆ab with g.∆ab , and then we have to collect all the coefficients (polynomials in the Pl¨ ucker coordinates of degree 2) of g.P , viewed as polynomial in the variables gi,j . For instance collecting the coefficients of the polynomial g.(∆35 ∆46 − ∆25 ∆56 ), ∆35 ∆46 − ∆25 ∆56 ∈ T ′ , we obtain 3495 polynomials that give some of the equations defining the Hilbert scheme 2 HilbP2 (K). 6. Generalization Let us prove now that equations obtained from Theorem 5.3 can be used to define the Hilbert scheme, as scheme representing the Hilbert functor (see for instance [16, Section 4.3]). Let us start recalling some relevant definitions and properties.

16


Definition 6.1 ([11]). Let X and Y be schemes and f : X → Y be a morphism of schemes. X is said to be flat over Y if OX is f -flat over Y i.e. for every x ∈ X, OX,x is a OY,f (x) -flat module. Definition 6.2 ([16]). The Hilbert functor of Pn relative to the polynomial p(t) is the contravarin ant functor HilbPp(t) : (schemes)◦ → Sets from the category of (schemes) to the category of Sets which associates to an object X of (schemes) the set   flat families Z ⊂ Pn × X of closed   n subschemes of Pn parametrized by X with HilbPp(t) (X) = (17)   fibers having Hilbert polynomial p(t) The Hilbert functor is representable and we will denote by Hilbnp(t) its representing scheme. n

n

Example 6.3. If X = Spec A, with A noetherian k-algebra, HilbPp(t) (A) = HilbPp(t) (Spec A) is given by the set of saturated homogeneous ideals I ⊂ A[x0 , . . . , xn ] such that Proj A[x0 , . . . , xn ]/I is flat over Spec A and for every prime ideal p ⊂ A, the Hilbert polynomial of the k(p)-graded algebra A[x0 , . . . , xn ]/I ⊗A k(p) is equal to p(t) where k(p) is the residue field Ap /pAp . Proposition 6.4. Let R = R0 ⊕ R1 ⊕ · · · be a graded ring such that R0 is a local noetherian ring and that R is finitely generated as R0 -algebra by elements of degree 1. Let M be a finitely generated graded R-module and let Mt be its component of degree t. Then, M is flat over R0 if and only if Mt is free over R0 for all t. Proof. See [6, Exercise 6.10].

Pn

Corollary 6.5. Let X = Spec A, where A is a noetherian k-algebra and a local ring. Hilbp(t) (A) is equal to the set of homogeneous saturated ideals I of A[x0 , . . . , xn ] such that A[x0 , . . . , xn ]t /It is a free A-module for all t > 0 and such that the rank of A[x0 , . . . , xn ]t /It is equal to p(t) for t ≫ 0. We recall two theorems by Gotzmann already introduced, but now stated in the more general case of coefficients in a local ring (see [7, 12]). Theorem 6.6 (Gotzmann’s Regularity Theorem). Let r be the Gotzmann number associated to the Hilbert polynomial p(t), A any noetherian ring and T = A[x0 , . . . , xn ]. Any homogeneous ideal I ⊂ T with Hilbert polynomial p(t) is r-regular, that is e − i)) = 0 Hi (Pn , I(r for i > 0, where Ie is the quasi-coherent sheaf associated to I.

Theorem 6.7 (Gotzmann’s Persistence Theorem). Let r be the Gotzmann number associated to the Hilbert polynomial p(t) and s > r an integer. Let A be any noetherian ring, T = A[x0 , . . . , xn ] and I a homogeneous ideal of T generated by Is . Set M = T /I, if Mt is a flat A-module of rank p(t) for t = s, s + 1, then Mt is flat of rank p(t) for all t > r.

Remark 6.8. By Theorem 6.6, we deduce that if I is a saturated homogeneous ideal of T with Hilbert polynomial p(t) whose Gotzmann number is r, then: (i) I>t = (It ), ∀ t > r; (ii) the value of Hilbert function of T /I coincides with the value of p(t) for each degree greater than or equal to r. Now we focus our attention on the case of an affine scheme X = Spec A, with A a noetherian k-algebra and a local ring with maximal ideal m and residue field K. Moreover we fix a Hilbert polynomial p(t) of degree d and Gotzmann number r. Applying Theorems 6.6 and 6.7, Remark 6.8, Corollary 6.5 and Nakayama’s Lemma, we deduce the following propositions.


17

n

Proposition 6.9. Given an integer s > r, HilbPp(t) (X) is in bijection with the subset W of p(s)

p(s+1)

GrSs (X) × GrSs+1 (X) defined by o n p(s+1) p(s) W = (Ts /Is , Ts+1 /Js+1 ) ∈ GrSs (X) × GrSs+1 (X) T1 · Is = Js+1 .

(18)

Remark 6.10. Keeping in mind Macaulay’s Estimate on the Growth of Ideals (see also [12, p(s+1) p(s) Corollary C.4]) and Nakayama’s Lemma, the subset W of GrSs (X) × GrSs+1 (X) is also equal to o n p(s+1) p(s) (19) W = (Ts /Is , Ts+1 /Js+1 ) ∈ GrSs (X) × GrSs+1 (X) T1 · Is ⊆ Js+1 . n

Proposition 6.11 ([12]). Given an integer s > r, HilbPp(t) (X) is in bijection with the subset G p(s)

of GrSs (X) given by o n p(s) G = Ts /Is ∈ GrSs (X) | Ts+1 /(T1 · Is ) is a free A-module of rank p(s + 1) . n

(20)

p(s)

Proposition 6.11 says that HilbPp(t) (X) can be viewed as a subset of GrSs (X) for s > r. Now we construct explicitly a set of equations that determine the Hilbert scheme Hilbnp(t) representing n

the functor HilbPp(t) as subscheme of the scheme Grp(r) (Sr ) representing the Grassmann functor p(r)

GrSr . Theorem 6.12. The Hilbert scheme Hilbnp(t) is a closed subscheme of the scheme Grp(r) (Sr ) p(r)

ucker representing the Grassmann functor GrSr and it can be defined by equations in the Pl¨ coordinates of degree smaller than or equal to d + 2. n

Proof. Theorem 5.3 proves the statement in the case X = Spec k, that is HilbPp(t) (k) is a closed p(r)

subscheme of GrSr (k) defined by equations of degree smaller than or equal to d + 2. Let us extend this result to the case of X = Spec A with A a noetherian k-algebra and a local ring with maximal ideal m and residue field K. p(r)

Step 1. Let Ir ∈ GrSr (X) be an A-submodule of Tr = Sr ⊗k A, that is Tr /Ir is a free A-module of rank p(r). Firstly let us prove that if equations given in Theorem 5.3 are satisfied, then Ir n p(r+1) belongs to HilbPp(t) (X) (i.e. T1 · Ir belongs to GrSr+1 (X) according to Proposition 6.11). p(r)

Let us consider Ir ∈ GrSr (X) satisfying equations of Theorem 5.3. Tensoring by the residue field K and using Nakayama’s Lemma, we can determine a free submodule Jr ⊂ Tr generated by q(r) monomials having the Borel-fixed property, such that maybe with a generic change of coordinates, the monomials N (Jr ) = {xβ ∈ Tr | xβ ∈ / Jr } form a basis of Tr /Ir as a free A-module (see [6]). Now we consider the exact sequence 0 → Ir → Tr → Tr /Ir → 0, and we tensor it by the residue field K, obtaining Ir ⊗k K → Sr (K) = K[x0 , . . . , xn ]r → Tr /Ir ⊗k K → 0. Called Ir (K) the image of Ir ⊗k K in Sr (K), by the assumptions and by Theorem 5.3 we n deduce that Ir (K) belongs to HilbPp(t) (K). Consequently, Jr (K) (resp. Jr ) also belongs to n n HilbPp(t) (K) (resp. HilbPp(t) (A)) and thus (Jr (K)) (resp. (Jr )) defines a Borel-fixed ideal with Hilbert polynomial p(t) in S(K) = K[x0 , . . . , xn ] (resp. in A[x0 , . . . , xn ]).

18


For a generic change of coordinates, as N (Jr ) is a rank q(r) with a basis of the form:   X xα − cαβ xβ  β x ∈N (Jr )

basis of Tr /Ir , Ir is also a free A-module of   xα ∈ Jr . 

(21)

Therefore we can choose a system of generators for T1 · Ir equal to that one described with the matrix Ar+1 in Proposition 5.1. Up to a change of coordinates, the A-module hD1 i generated by the lines of D1 (and by definition equal to x0 Ir + . . . + xd Ir ) contains a family F of q ′ (r + 1) polynomials of the form:     X F = xγ − cγη xη xγ ∈ x0 Jr + . . . + xd Jr . (22)   η x ∈N (Jr+1 )

Let us prove it by induction on 0 6 i 6 d for xα ∈ xi Jr . Let xγ = x0 xα with xα in Jr . Among the generators (21) of Ir there is X xα − cαβ xβ , xβ ∈N (Jr )

thus x0 xα −

X

cαβ x0 xβ

xβ ∈N (Jr )

belongs to x0 Ir ⊂ x0 Ir + . . . + xd Ir . Moreover because of (Jr ) is Borel-fixed and (Jr+1 : S1 ) = Jr n (Jr ∈ HilbPp(t) (A)), it is easy to check that x0 N (Jr ) ⊂ N (Jr+1 ). Hence the assertion is proved for i = 0 and let us suppose that it holds for all 0 6 j < i. Considered xγ = xi xα with xα ∈ Jr , again X xi xα − cαβ xi xβ ∈ xi Ir ⊂ x0 Ir + . . . + xd Ir . xβ ∈N (Jr )

If xi xβ (xβ ∈ N (Jr )) does not belong to N (Jr+1 ), then there exists xǫ ∈ Jr such that xi xβ = xj xǫ and j < i because of the Borel-fixed property. Then, by induction, we can replace xi xβ = xj xǫ with an element of the A-module generated by N (Jr+1 ) modulo x0 Ir + . . . + xd Ir , finally proving that the family F described in (22) belongs to hD1 i (i.e. to x0 Ir + . . . + xd Ir ). As equations of Theorem 5.3 are satisfied, equations (14) and (15) are also satisfied for a generic change of coordinates, so that we can assume without loss of generality that there exist Jr Borel-fixed and F as in (22), such that Ir satisfies (14) and (15). Now we want to show that equations (14) and (15) imply that F generates the A-module hD2 i spanned by the lines of D2 . As a matter of fact, equations (14) and (15) imply that the exterior product between q ′ (r + 1) polynomials in hD1 i and one polynomial in hD2 i always vanishes. In particular, the exterior product between the q ′ (r + 1) polynomials that belong to F and any polynomial g in hD2 i is equal to zero. We deduce easily that g belongs to hFi and that F generates hD2 i. Moreover F generates hD1 i. With the same reasoning used in the proof of Theorem 5.3 and in Remark 5.4, it is easy to prove that any exterior product between q ′ (r + 1) + 1 polynomials in hD1 i is equal to zero. In particular, the exterior product between the q ′ (r + 1) polynomials that belong to F and any polynomial g in hD1 i is equal to zero. So again g belongs to hFi and F generates hD1 i (keeping in mind that the free A-module Tr has a basis that contains F).


19

Finally, we conclude that Ir+1 is a free A-module with basis F plus the polynomials represented by the lines in the first rows of Ar+1 and rewriting this family of polynomials using linear combinations of elements in F we can obtain a basis of the form     X γ η γ x − cγη x x ∈ Jr+1 . (23)   η x ∈N (Jr+1 )

n

Tr+1 /Ir+1 turns out to be an A-module with basis N (Jr+1 ), so Ir ∈ HilbPp(t) (X). p(r)

n

Step 2. Let us suppose that Ir ∈ GrSr (X) belongs to HilbPp(t) (X) and let us prove that it satisfies equations given in Theorem 5.3. This is equivalent to prove that equations (14) and (15) are satisfied for a generic changes of coordinates. From Galligo’s Theorem [6, Theorem 15.20] and Nakayama’s Lemma, there exists a Borel-fixed monomial ideal J with Hilbert polynomial p(t), such that for a generic changes of coordinates, N (Jr ) and N (Jr+1 ) are a basis of respectively Tr /Ir and Tr+1 /Ir+1 as free A-modules. As mentioned in Step 1, we can represent Ir+1 with the matrix Ar+1 introduced in Proposition 5.1 and find a family F in hD1 i of the form (22). As N (Jr+1 ) is a basis of Tr+1 /Ir+1 as a free A-module, every polynomial given by a line in D1 or D2 belongs to hFi. Therefore equations (14) and (15) are satisfied for a generic change of coordinates and equations of Theorem 5.3 are satisfied. Step 3. From the local approach given by Proposition 6.9, the following isomorphism holds   p(r)   0 → Ir → Sr ⊗k OX → E → 0 ∈ GrSr (X)   Pn such that (24) Hilbp(t) (X) ≃    0→S ·I →S ′ → 0 ∈ Grp(r+1) (X)  ⊗ O → E 1 r r+1 k X Sr+1 n

which induces a natural transformation from the Hilbert functor HilbPp(t) to the Grassmann p(r)

functor GrSr . Using the equivalence proved in Step 1 and Step 2 in a local approach, we deduce   p(r)   Sr ⊗k OX → E → 0 ∈ GrSr (X)   Pn Hilbp(t) (X) ≃ (25) such that   0 p(r)  {∆I }  E) satisfy equations of Theorem 5.3 I={i1 ,...,ip(r) } ∈ H (X, ∧

where the element ∆I ∈ H 0 (X, ∧p(r) E) associated to the multi-index I = {i1 , . . . , ip(r) } is given by the image of xα(i1 ) ∧· · ·∧xα(ip(r) ) ∈ H 0 (X, ∧p(r) Sr ⊗k OX ) into H 0 (X, ∧p(r) E) using the p(r)-th exterior power of the morphism: Sr ⊗k OX → E → 0. p(r)

ucker’s relations, we have Using the representation of the Grassmann functor GrSr via the Pl¨   Sr ⊗k OX → E → 0 such that   n ∧p(r) E is an invertible sheaf and {∆I }I={i1 ,...,ip(r) } ∈ H 0 (X, ∧p(r) E) HilbPp(t) (X) ≃ (26)   satisfy equations of Theorem 5.3 and Pl¨ ucker’s relations Finally, because of the isomorphism

n HilbPp(t) (X) ≃ Hom X, Proj k[∧p(r) Sr ]/H ,

ucker’s (see [11, Theorem 7.1]) where H is the homogeneous ideal of k[∧p(r) Sr ] generated by Pl¨ relations and equations of Theorem 5.3, by definition, the Hilbert scheme is equal to Proj k[∧p(r) Sr ]/H.

20


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