The close relation between border and Pommaret marked bases

Given a finite order ideal $\mathcal O$ in the polynomial ring $K[x_1,\dots, x_n]$ over a field $K$, let $\partial \mathcal O$ be the border of $\mathcal O$ and $\mathcal P_{\mathcal O}$ the Pommaret basis of the ideal generated by the terms outside $\mathcal O$. In the framework of reduction structures introduced by Ceria, Mora, Roggero in 2019, we investigate relations among $\partial\mathcal O$-marked sets (resp.~bases) and $\mathcal P_{\mathcal O}$-marked sets (resp.~bases). We prove that a $\partial\mathcal O$-marked set $B$ is a marked basis if and only if the $\mathcal P_{\mathcal O}$-marked set $P$ contained in $B$ is a marked basis and generates the same ideal as $B$. Using a functorial description of these marked bases, as a byproduct we obtain that the affine schemes respectively parameterizing $\partial\mathcal O$-marked bases and $\mathcal P_{\mathcal O}$-marked bases are isomorphic. We are able to describe this isomorphism as a projection that can be explicitly constructed without the use of Gr\"obner elimination techniques. Several examples are given along all the paper.


Introduction
Let K be a field and R A := A[x 1 , . . . , x n ] the polynomial ring over a Noetherian K-algebra A in the variables x 1 < · · · < x n . Let T denote the set of terms, i.e. monic monomials, in R A .
Given an order ideal O ⊆ T, the ideals I ⊂ R A such that O is an A-basis of R A /I have been extensively investigated and characterized in literature, for instance because they are suitable tools for the study of Hilbert schemes. These ideals can be identified by means of particular sets of generators called marked bases.
If O is a finite order ideal, then also the border ∂O is finite and the ideal generated by the terms outside O admit a Pommaret basis P O , which is contained in ∂O. Hence, in this paper we consider finite order ideals and focus on the known characterizations of the ideals I ⊂ R A , such that O is an A-basis of R A /I, that have been obtained by using either border bases (∂O-marked bases in this paper) or marked bases over a quasi-stable ideal (P O -marked bases in this paper). We explicitly describe a close relation between these types of marked bases.
Given a finite order ideal O, ∂O-marked sets and bases are made of monic marked polynomials whose head terms form ∂O. Border bases were first introduced in [17], in the context of Gröbner bases, for computing a minimal basis of an ideal of polynomials vanishing at a set of rational points, also using duality. Border bases were then investigated from a numerical point of view because of their stability with respect to perturbation of the coefficients [20] and have also attracted interest from an algebraic point of view (see [12], [13,Section 6.4]). Furthermore, given a finite order ideal O, the ∂O-marked bases parameterize an open subset of a punctual Hilbert scheme (e.g. [10,15]). This fact was used, for instance, to investigate elementary components of punctual Hilbert schemes in [11].
Recall that every Artinian monomial ideal in R K has a Pommaret basis. So, given a finite order ideal O, we can consider the Pommaret basis P O of the monomial ideal generated by T \ O and construct P O -marked sets and bases, i.e. sets and bases made of monic marked polynomials whose head terms form P O . For every strongly stable ideal, P O -marked bases were first introduced in [9] and investigated in [6] with the aim to parameterize open subsets of a Hilbert scheme and study it locally. We highlight that P O -marked bases do not need any finiteness assumption on the underlying order ideal. They were considered in [3,7] in the case of homogeneous polynomials. In [4] non-homogeneous P O -marked bases were studied, obtaining more efficient computational techniques than those in the homogeneous case. An affine scheme parameterizing P O -marked bases has been already described and used, for instance in [2,5], to successfully investigate Hilbert schemes.
The goal of our work is comparing ∂O-marked sets (and bases) and P O -marked sets (and bases) for a given finite order ideal O. We use the framework of reduction structures introduced in [8] and a functorial approach in order to compare the schemes parameterizing these two different types of bases. We prove that there is a close relation among ∂O-marked bases and P O -marked bases (see Theorem 3.6) and observe that the affine schemes parameterizing the ideals generated by these two types of bases are isomorphic, also giving an explicit isomorphism (see Corollary 3.9, Theorem 4.1 and Corollary 4. 3). Several examples are exhibited along all the paper.
The paper is organized in the following way.
In Section 1 we recall some general notations and facts, the framework of reduction structures introduced in [8] and some known results for the Pommaret reduction structure.
In Section 2 we focus on ∂O-marked bases. Observing that a set B of marked polynomials on ∂O always contains a set P of marked polynomials on P O , we prove that B is a ∂O-marked basis if and only if P is a P O -marked basis and generates the same ideal as B (Theorem 2.3). We also compare the border reduction structure implicitly considered in [13] with the one we give in Definition 2.1. In particular, we relate the border division algorithm of [13,Proposition 6.4.11] to the reduction relation induced by the border reduction structure, where the terms of the border are ordered according to the degree. In this setting, we can prove a Buchberger's criterion for ∂O-marked bases (Proposition 2.12), which is alternative to that of [13,Proposition 6.4.34].
In Section 3, using a functorial approach, we easily prove that the scheme parameterizing ∂Omarked bases, called border marked scheme, and the scheme parameterizing P O -marked bases, called Pommaret marked scheme, are isomorphic (Corollary 3.9). The monicity of the marked sets we consider is crucial for the use of functors. In fact, although in [8] the authors deal with the polynomial ring R K , everything works also in R A thanks to the monicity of marked polynomials and sets.
In Section 4, we explicitly exhibit an isomorphism between the border marked scheme and the Pommaret marked scheme using Theorem 2.3. As a byproduct, we prove that there is always a subset of the variables involved in the generators of the ideal defining the border marked scheme that can be eliminated (see Corollary 4.2), obtaining in this way the ideal defining the Pommaret marked scheme. This elimination does not use Gröbner elimination techniques, which are in theory useful, but practically impossible to use (see Example 4.5). Corollary 4.3 is the geometric version of Corollary 4.2.

Generalities and setting
Let K be a field, R := K[x 1 , . . . , x n ] the polynomial ring in n variables with coefficients in K, T the set of terms in R. If Y is a subset of {x 1 , . . . , x n }, we denote by T[Y ] the subset of T containing only terms in the variables in Y . We denote by A a Noetherian K-algebra with unit 1 K and set R A := A ⊗ K R. When it is needed, we assume x 1 < · · · < x n (this is just an ordering on the variables, this is not a term order). For every term τ in T we denote by min(τ ) the smallest variable appearing in τ with non-null exponent.
Definition 1.1. A monomial ideal J ⊂ R is quasi-stable if, for every term τ ∈ J and for every x i > min(τ ), there is a positive integer s such that x s i τ / min(τ ) belongs to J. It is well known that a quasi-stable ideal J has a special monomial generating set that is called Pommaret basis and has a very important role in this paper. The terms in the Pommaret basis of J can be easily detected thanks to the following property (see [7,Definition 4.1 and Proposition 4.7]). For every term σ ∈ J, (1.1) σ belongs to the Pommaret basis of J ⇔ σ min(σ) / ∈ J.
Given a monomial ideal J ⊆ R, the set of terms outside J is an order ideal. On the other hand, if O is an order ideal, then J is the monomial ideal such that J ∩ T = T \ O.
Along the paper we only consider finite order ideals O. This is equivalent to the fact that the Krull dimension of the quotient ring R/J is zero, i.e R/J is Artinian. In this case, J is quasi-stable [ Since O ∪ ∂O is an order ideal too, for every integer k ≥ 0 we can also define the k-th border ∂ k O of O in the following way: For every finite order ideal O ⊂ T, it is immediate to observe that P O is contained in ∂O, thanks to (1.1).
Given a subset T ⊆ T of terms, we denote by T A the module generated by T over A. Moreover, for every term α, the set C T (α) := {τ α | τ ∈ T } is called cone of α by T . • H ⊆ T is a finite set of terms; • for every α ∈ H, T α ⊆ T is an order ideal, called the multiplicative set of α, such that ∪ α∈H C Tα (α) = (H); • for every α ∈ H, L α is a finite subset of T \ C Tα (α) called the tail set of α. Given a reduction structure J = (H, L, T ), we say that: J has maximal cones if, for every α ∈ H, T α = T; J has disjoint cones if, for every α, Recall that the support of a polynomial f ∈ R A is the finite subset supp(f ) ⊆ T of those terms that appear with non-null coefficients in f . Definition 1.6. [19] A marked polynomial is a polynomial f ∈ R A together with a specified term of supp(f ) which appears in f with coefficient 1 K . It will be called head term of f and denoted by Ht(f ). Let O H be the order ideal given by the terms of T outside the ideal generated by H. An H-marked set F is called an H-marked basis if O H is a free set of generators for R A /(F ) as Given a reduction structure J = (H, L, T ) and an H-marked set F , a (H)-reduced form modulo (F ) of a polynomial g ∈ R A is a polynomial h ∈ R A such that g − h belongs to (F ) and supp(h) ⊆ O H . If there exists a unique (H)-reduced form modulo (F ) of g then it is called (H)-normal form modulo (F ) of g and is denoted by Nf(g).
In the following, when O H is finite, we will write O H -reduced form (resp. O H -normal form) instead of (H)-reduced form (resp. (H)-normal form). Furthermore, for any order ideal O, if h belongs to O A , then we say that h is O-reduced. Remark 1.8. Given a reduction structure J = (H, L, T ), if F is an H-marked basis, then every polynomial g ∈ R A admits the (H)-normal form Nf(g) modulo (F ). This is a direct consequence of the fact that R A decomposes in the direct sum (F ) ⊕ O H A , so that for every g ∈ R A there is a unique writing The definition of a reduction relation over polynomials can be useful to computationally detect reduced and normal forms. Definition 1.9. [8, page 105] Given a reduction structure J = (H, L, T ) and an H-marked set F , the reduction relation associated to F is the transitive closure → + F J of the relation on R A that is defined in the following way. For g, h ∈ R A , we say that g is in relation with h and write g → F J h if there are terms γ ∈ supp(g) and α ∈ H such that γ = αη belongs to C Tα (α) and Given a reduction structure J = (H, L, T ) and an H-marked set F , the reduction relation → + F J is Noetherian if there is no infinite reduction chain g 1 → F J g 2 → F J . . . . The reduction structure J is said Noetherian if → + F J is Noetherian for every H-marked set F (see [8,Section 5]).
Moreover, → + F J is confluent if for every polynomial g ∈ R A there exists only one (H)-reduced form h modulo (F ) such that g → + F J h. The reduction structure J is confluent if → + F J is confluent for every H-marked set F (see [8,Definition 7.1]).

Proposition 1.14. (Buchberger's criterion for Pommaret marked bases) [4, Proposition 5.6]
Let O be a finite order ideal, consider the Pommaret reduction structure J P O and let P be a P O -marked set. The following are equivalent: (1) P is a P O -marked basis; (2) for every p, p ′ ∈ P such that (Ht(p), Ht(p ′ )) is a non-multiplicative couple, S(p, p ′ ) → + P J P O 0.

Border reduction structure
In this section, given a finite order ideal O, we focus on reduction structures such that H is the border of O and their relations with the Pommaret reduction structure. Table 1 and Lemma 13.2] Given a finite order ideal O, let the terms of the border ∂O be ordered in an arbitrary way and labeled coherently, i.e. for every The border reduction structure J ∂O := (H, L, T ) is the reduction structure with H = ∂O and, for every The border reduction structure J ∂O given in Definition 2.1 is a substructure of J ′ , because the multiplicative sets of J ∂O are contained in those of J ′ (see [8,Definition 3.4] for the definition of substructure).
Like observed in [8,Remark 4.6], the definition of marked basis only depends on the ideal generated by the marked set we are considering, and this notion does not rely on the reduction relation associated to the reduction structure we are considering. However, in order to prove that the notions of ∂O-marked basis and P O -marked basis associated to J ∂O and J P O , respectively, are closely related, we need the features of the reduction relation given by the Pommaret reduction structure. Theorem 2.3. Let O be a finite order ideal, J ∂O be the border reduction structure, with terms of ∂O ordered arbitrarily, and J P O be the Pommaret reduction structure. Let B be a ∂O-marked set in R A , P the P O -marked set contained in B and B ′ = B \ P . Then, B is a ∂O-marked basis ⇔ P is a P O -marked basis and B ′ ⊂ (P ).
Since the P O -marked set P is a subset of B, we also have (P ) ∩ O A = {0}. Furthermore, since the Pommaret reduction structure is Noetherian and confluent, every polynomial in R A has a O-reduced form modulo (P ), that is (P ) + O A = R A by Remark 1.10. These two facts together imply that P is a marked basis.
. This means thatb ′ is also the only O-normal form modulo (B), which is equal to 0 by hypothesis (see Remark 1.8). Hence, b ′ belongs to (P ).
Assume now that P is a P O -marked basis and B ′ ⊂ (P ). Observing that in the present hypotheses (P ) ⊕ O A = R A and (P ) = (P ∪ B ′ ) = (B), we immediately conclude.
as in Example 1.4, the ∂O-marked set B e 1 ,e 2 given by the following polynomials For every e 1 , e 2 ∈ K, P is a P O -marked basis, by Proposition 1.14. If e 1 = 3 or e 2 = 0, then b 4 does not belong to (P ), hence, by Theorem 2.3, B e 1 ,e 2 is not a ∂O-marked basis for these values of e 1 and e 2 . If e 1 = 3 and e 2 = 0 then b 4 belongs to (P ), hence B 3,0 is a ∂O-marked basis.
In [13], the authors consider a procedure called border rewrite relation [13, page 432] which is the reduction relation of the reduction structure J ′ presented in Remark 2.2. The reduction structure J ′ is not Noetherian, as highlighted in [13,Example 6.4.26]. This is due to the fact that J ′ has maximal cones and, in general, there is no term order with respect to which, for every f in a ∂O-marked set F and for every γ in O, the head term Ht(f ) of f is bigger than γ (see [19] and [8,Theorem 5.10]).
In general, given a reduction structure J , disjoint cones are not sufficient to ensure Noetherianity. For instance, also the substructure J ∂O of J ′ is in general not Noetherian. We highlight this rephrasing the above quoted example of [13] for the reduction structure J ∂O .
We consider the border reduction structure J ∂O with the terms of the border ordered in the following way: The reduction structure J ∂O has disjoint cones, however this does not imply Noetherianity. Indeed, consider the term γ = x 2 1 x 2 2 . When we reduce γ by → B J ∂O , we fall in the same infinite loop as using the border rewrite relation of [13]. Recalling that a term γ ∈ (∂O) is reduced using the term β i with i = max{j | β j ∈ ∂O divides γ}, we obtain In the following, we will order the terms of ∂O either increasingly by degree (terms of the same degree are ordered arbitrarily), or increasingly according to a term order ≺. In the first case, we will denote the reduction structure with J ∂O,deg and, in the second case, with J ∂O,≺ .
For every term order ≺, the reduction structure J ∂O,≺ is Noetherian [8, page 127]. We now focus on properties of the reduction relation of a border reduction structure J ∂O,deg and explicitly prove that J ∂O,deg is Noetherian and confluent. We use properties of the border described in [13,Section 6.4.A] and results in [8].
Lemma 2.6. Let O be a finite order ideal.
(1) If γ is a term in (T \ O) and β ∈ ∂O is a term of maximal degree dividing γ, then Proof. Item (1)  Proposition 2.7. Given a finite order ideal O, consider the border reduction structure J ∂O,deg and a ∂O-marked set B. Then, the reduction relation → + B J ∂O,deg is Noetherian and confluent. Proof. First we prove that J ∂O,deg has disjoint cones, that is C T β i (β i ) ∩ C T β j (β j ) = ∅, for every β i , β j ∈ ∂O. Indeed, assume j > i and consider γ ∈ C T β i (β i ). By definition of cone, there is η ∈ T β i such that ηβ i = γ and for every j > i, β j does not divide γ. Hence, γ does not belong to C T β j (β j ). Furthermore, J ∂O,deg is Noetherian. It is sufficient to consider T with the well founded order given by the index of a term with respect to O, and apply [8,Theorem 5.9]. Indeed, consider f, g ∈ R A such that f → B J ∂O,deg g and let γ = ηβ i ∈ supp(f ) ∩ C T β i (β i ) be the term which is reduced, i.e. f − cηb i = g, with c ∈ R A the coefficient of γ in f and b i the polynomial of B such that Ht(b i ) = β i . Consider a term γ ′ in supp(g) \ supp(f ). By construction we obtain that γ ′ is divisible by η. So, if γ ′ belongs to C T β j (β j ) for some β j ∈ ∂O, then by Lemma 2.6 We highlight that, given the border reduction structure J ∂O,deg and a ∂O-marked set B, the reduction relation → + F J ∂O,deg is equivalent to the border division algorithm of [13,Proposition 6.4.11], in the sense that the O-reduced forms obtained by the reduction relation → + F J ∂O,deg are normal O-reminders of the border division algorithm (see [13, page 426]).
Recall that for a given finite order ideal O, every ∂O-marked set contains a P O -marked set. Nevertheless, the different reduction relations applied on the same polynomial in general give different O-reduced forms.
, as in Example 1.4, and the border reduction structure J ∂O,deg and the Pommaret reduction structure J P O . Let B ⊂ R be the ∂O-marked set given by the following polynomials: We also compute an O-reduced form modulo (P ) in the following way: We now focus on how the reduction relations corresponding to border reduction structures are used in order to obtain marked bases. Definition 2.9. For every α, α ′ terms in a finite set M ⊂ T, (α, α ′ ) is a neighbour couple if either α = x j α ′ for some variable x j or x i α = x j α ′ for some couple of variables (x i , x j ).
The definition of neighbour couple is due to [13, Definition 6.4.33 c)] in the framework of border bases, however Definition 2.9 is given for any finite set of terms, not necessarily the border of an order ideal. Furthermore, the notion of neighbour couple does not depend on the reduction structure we are considering.
Given a finite order ideal O and recalling Definition 1.13, if we consider the Pommaret reduction structure J P O , then a couple of terms in P O can be simultaneously non-multiplicative and neighbour, although this is not always the case (see Example 2.14).
We now investigate the non-multiplicative couples for the border reduction structure J ∂O,deg , showing that non-multiplicative couples are always neighbour couples in this case. Lemma 2.10. Given a finite order ideal O, consider the border reduction structure J ∂O,deg , and let (β i , β j ) be a non-multiplicative couple of ∂O, i.e. x ℓ β i = δβ j for some x ℓ / ∈ T β i and δ ∈ T β j . Then j > i and deg(δ) ≤ 1.
Proof. By definition of T β j for a border reduction structure (Definition 2.1), the equality x ℓ β i = δβ j , with x l / ∈ T β i and δ ∈ T β j , implies that j > i. Furthermore deg(β i ) ≤ deg(β j ), since in J ∂O,deg the terms of δO are ordered according to increasing degree. This implies that deg(δ) ≤ deg(x ℓ ) = 1.
As already observed in Remark 2.2, J ∂O,deg is a substructure of J ′ . So, we can rephrase Buchberger's criterion for border marked bases given in [13,Proposition 6.4.34] in terms of the reduction relation given by J ∂O,deg . (1) B is a ∂O-marked basis; (2) for every ( Applying [8,Theorem 11.6] to the reduction structure J ∂O,≺ , for some term order ≺, an alternative Buchberger's criterion can be obtained. However, this criterion might involve couples of terms in ∂O which are neither neighbour, nor non-multiplicative, unless J ∂O,≺ has multiplicative variables. For instance, if ≺ lex is the lex term order, then the reduction structure J ∂O,≺ lex has multiplicative variables (see [8,Theorem 13.5]).
In general, the reduction structure J ∂O,deg does not have multiplicative variables, not even if we consider J ∂O,≺ for a degree compatible term order [8,Example 13.4]. Nevertheless, we now show that it is sufficient to consider non-multiplicative couples of terms ∂O in J ∂O,deg in order to obtain another Buchberger's criterion for border bases. Proof. Here, we prove the equivalence between Proposition 2.11(2) and item (2). By Lemma 2.10, one implication is immediate, so we move to proving that item (2) implies Proposition 2.11 (2). Consider now a neighbour couple (β i , β j ) which is not a non-multiplicative couple. Since ∂O is increasingly ordered by degree, if x s β i = β j then j > i, hence x s / ∈ T β i by definition of T as given in Definition 2.1. Furthermore, if x s β i = x l β j , then it is not possible that x s belongs to T β i and simultaneously x ℓ ∈ T β j , again by Definition 2.1.
Hence, the only case of neighbour couple which is not non-multiplicative is x s β i = x l β j with x s / ∈ T β i and x ℓ / ∈ T β j . Let β k be the unique term in ∂O such that x s β i = x ℓ β j = δβ k with δ ∈ T β k . Hence both (β i , β k ) and (β j , β k ) are non-multiplicative couples and by Lemma 2.10 we obtain that i, j < k and deg(δ) Remark 2.13. The most used effective criterion to check whether a ∂O-marked set is a basis is not a Buchberger's criterion. The commutativity of formal multiplication matrices is usually preferred (see [18,Theorem 3.1]). This is simply due to the fact that the border reduction given in [13] is neither Noetherian nor confluent, hence when in a statement one finds "f → + B J ∂O 0", one should read "it is possible to find a border reduction path leading f to 0".
In this case the reduction structure J ∂O,deg has multiplicative variables.
Similarly to what is done in [20,Definition 8.5], in Figures 1, 2, 3 we represent graphs whose vertices are the terms in ∂O (we use bullets for the terms in P O and a star for the unique term in ∂O \ P O ), and whose edges are given by either neighbour couples of terms in ∂O (Figure 1), or non-multiplicative couples of terms ∂O in J ∂O,deg (Figure 2) or non-multiplicative couples of terms P O in J P O (Figure 3). In Figures 2 and 3, we use arrows for edges; the arrow starts from b i and ends at and , for every t ≥ 2, (b it , b i t+1 ) are neighbour couples of terms in ∂O of J ∂O,deg . In Example 2.14, this is the case for the non-multiplicative couples (b 6 , b 5 ) and (b 7 , b 5 ) of terms in P O of J P O . This means that in general there are less S-polynomials to consider in Proposition 1.14(2) than those in Proposition 2.11 (2) or Proposition 2.12(2).

Border and Pommaret marked functors and their representing schemes
From now, for what concerns border reduction structures, given a finite order ideal O, we only consider the border reduction structure J ∂O,deg , which will be simply denoted by J ∂O .
Given a finite order ideal O, the goal of this section is proving that the two affine schemes, which respectively parameterize ∂O-marked bases and P O -marked bases, are isomorphic.  We now investigate Mb J ∂O and Mb J P O more in detail in order to understand the relation among them.

Observe that the generic H-marked set is a H-marked set in R
The generic ∂O-marked set was first introduced in [14, Definition 3.1]. We denote it by B, and observe that, up to relabelling the parameters C, B always contains the generic P O -marked set, that we denote by P. From now on we denote by B ′ the set B \ P and byC the set of parameters in the polynomials of B ′ . Example 3.4. We consider again the finite order ideal O = {1, x 1 , x 2 , x 1 x 2 }. Then, the generic ∂O-marked set B is given by the following polynomials: The border marked scheme was introduced in [14], and it was further investigated in [10,15]. In [15] the interested reader can find a different proof of the fact that Mb J ∂O is the functor of points of the border marked scheme. In [14, Definition 3.1], although the functor is not explicitly defined, the authors define the border marked scheme using the commutativity of formal matrices.
For the sake of completeness, we explicitly prove that the affine scheme defined by the ideal B of Definition 3.7 represents the functor Mb J ∂O just like the schemes in [14, Definition 3.1], [15,Proposition 3]. Our proof is inspired by [8,Appendix A]. The underlying idea of the proof is to apply Buchberger's criterion (Proposition 2.11) to the generic ∂O-marked set, in order to define a set of conditions to impose on these parameters for obtaining a ∂O-marked basis.

Proof. For every K-algebra A, there is a 1-1 correspondence between Hom(K[C]/B, A) and
Mb J ∂O (A). In fact, on the one hand we consider a morphism of K-algebras π : K[C]/B → A and extend it in the obvious way to a morphism between the polynomial rings R K[C]/B and R A . Then, we can associate to every such morphism π the ∂O-marked basis π(B), where B is the generic ∂O-marked set. The ideal generated by π(B) belongs to Mb J ∂O (A).
On the other hand, every ∂O-marked basis B = {b i = τ i − c ij σ j |c ij ∈ A} i=1,...,m ⊂ R A generates an ideal belonging to Mb J ∂O (A) and defines a morphism π B : K[C]/B → A given by π B (C i,j ) = c i,j , thanks to Proposition 2.11. This 1-1 correspondence commutes with the extension of scalars, because for every morphism φ : A → A ′ and for every ∂O-marked basis B we have φ( The functorial approach combined with Theorem 2.3 gives the following result.
They are obtained computing the O-reduced forms modulo B by → + B ∂O of the S-polynomials of the neighbour couples The ideal P ⊂ K[C \C] of Definition 3.5 is generated by the following polynomials Observe that among the generators of B, there are polynomials containing C 4,i , i = 1, . . . , 4. These variables do not appear in the generators of P, because the polynomial b 4 is not involved in its construction.
In [4] and in [6] (in the latter case, under the hypothesis that K has characteristic 0), the authors present an open cover for the Hilbert scheme parameterizing d-dimensional schemes in P n K with a prescribed Hilbert polynomial; the open subsets are Pommaret marked schemes, but in order to cover the whole Hilbert scheme, the action of the general linear group GL K (n + 1) is needed. We now highlight that, for the Hilbert scheme parameterizing 0-dimensional subschemes of length m in A n K , the Pommaret marked schemes cover the whole Hilbert scheme without any group action. In the next statement, Proposition 3.11, we also recall the known analogous result for border marked schemes, see [14, Remark 3.

The isomorphism between the border and Pommaret marked schemes
In the present section we explicitly present an isomorphism between the border and the Pommaret marked schemes of the same finite order ideal O. We describe the algebraic relation between the ideals B ⊂ K[C] and P ⊂ K[C \C] defining the two schemes and, as a byproduct, obtain a constructive method to eliminate the variablesC. Proof. Consider the generic ∂O-marked set B and the generic P O -marked set P contained in B. Let B ′ = {b ′ i } be the set of marked polynomials in B \ P and denote byC the parameters appearing in the polynomials in B ′ . Thanks to Theorem 2.3, we can obtain the ideal B defining the border marked scheme in the following alternative way with respect to Definition 3.7.
Consider τ ′ i ∈ ∂O \ P O and let b ′ i be the marked polynomial in B such that Ht(b ′ i ) = τ ′ i . By the Pommaret reduction relation, compute the unique O-reduced polynomial h ′ i such that . By Theorem 2.3, the following equality of ideals in K[C] holds:  Let O be a finite order ideal. There is a projection morphism from A |C| to A |C\C| such that its restriction to the border marked scheme is an isomorphism between the border marked scheme and the Pommaret one. Proof. With the notation introduced in the proof of Theorem 4.1, let Φ be the K-algebra sur- The morphism Φ corresponds to the morphism φ : A |C\C| → A |C| , which associates to a closed K-point P := (a 1 , . . . , a |C\C| ) the closed K-point Q := (a 1 , . . . , a |C\C| , q i,j (a 1 , . . . , a |C\C| )). The kernel of Φ coincides with the ideal B ′ generated by the polynomials C i,j − q i,j , as C i,j varies inC. Hence, from Φ we obtain a K-algebra isomorphismΦ : K[C]/B ′ → K[C \C] which corresponds to an isomorphismφ : A |C\C| = Spec (K[C \C]) → Spec (K[C]/B ′ ) that is the inverse of the restriction to Spec (K[C]/B ′ ) of the projection from A |C| to A |C\C| .
Since Φ(B) = P, from Φ we obtain a K-algebra isomorphism Ψ : K[C]/B → K[C \C]/P, which gives to the inverse of the desired projection from Spec (K[C]/B) to Spec (K[C \C]/P).
Corollary 4.2 states that it is possible to know a priori a subset of the variables in C that can be eliminated from the border marked scheme, and also gives in its proof an effective method which does not use any Gröbner elimination computation. In Example 4.4, the polynomials generating B ′ that allow this elimination are also generators of B (see Example 3.10). This is not the case in general as the following example shows, although in the generators of B there are often linear terms that allow the elimination of some variables in C. For the interested reader, ancillary files containing the generic marked sets B and P, the set of eliminable parametersC and the generators of the ideals B, P and B ′ are available at https://sites.google.com/view/cristinabertone/ancillary/borderpommaret Constructing B as in Definition 3.7 and thanks to Theorem 3.8, the border marked scheme of O is a closed subscheme of A |C| = A 84 , and its defining ideal is generated by a set of 126 polynomials of degree 2.
Constructing P as in Definition 3.5 and thanks to Theorem 3.6, the Pommaret marked scheme of O is a closed subscheme of A |C\C| = A 42 , and its defining ideal is defined by a set of 56 polynomials of degree at most 5.
The polynomials C ij − q ij , which generates the ideal B ′ and allow the elimination of the 42 parametersC, have degrees between 2 and 4. In particular, there are 14 polynomials of degree 2 that generate B ′ and also belong to the set of polynomials given in Definition 3.7 generating B. Hence, 14 of the 42 variables inC can be eliminated by some generators of B. Then among the polynomials generating B ′ there are 14 polynomials of degree 3 and 14 polynomials of degree 4, which do not belong to the set of the generators of B we consider, which have degree 2.
So, Corollary 4.2 has two important consequences on the elimination of variables from the border marked scheme. Firstly, it identifiesC as a set of eliminable variables. In the present example, if one only knows that 42 of the variables in C are eliminable, it is almost impossible to find an eliminable set of this size without knowing that such a set is given by the variables appearing in marked polynomials with head terms in ∂O \ P O . Secondly, even if one knows that the variables inC can be eliminated, this is not feasible by a Gröbner elimination. We stopped the computation of such a Gröbner basis on this example after one hour of computation by CoCoA 5 (see [1]) and Maple 18 (see [16]), while it takes few seconds to compute the polynomials generating B ′ by → + P J P O .