Canonical Hilbert-Burch matrices for power series
aa r X i v : . [ m a t h . A C ] A p r CANONICAL HILBERT-BURCH MATRICES FOR POWER SERIES
ROSER HOMS AND ANNA-LENA WINZ
Abstract.
We give a parametrization of zero-dimensional ideals in the power se-ries ring k [[ x, y ]] with a given leading term ideal with respect to local lex ordering τ in terms of certain canonical Hilbert-Burch matrices. This is an extension tothe local setting of the parametrizations of Gr¨obner cells obtained in the polyno-mial ring k [ x, y ] by Conca-Valla in [CV08] and Constantinescu in [Con11] for thelexicographic and degree-lexicographical orderings, respectively. Contents
1. Parametrization of ideals in k [ x, y ] 32. From polynomials to power series 52.1. Enhanced standard basis and Grauert’s division 52.2. Lifting of syzygies in local rings 73. Towards a parametrization of ideals in k [[ x, y ]] 94. Parametrization for lex-segment leading term ideals 145. Applications to the construction of Gorenstein rings 21Acknowledgements 23References 23Punctual Hilbert schemes Hilb d ( k [[ x , . . . , x n ]]) parametrize points of multiplicity d at the origin. Its counterpart in commutative algebra is the study of local Artinian k -algebras of multiplicity d . In [Poo08], Poonen provides a complete classification of k -algebras with d ≤ k . For d ≥
7, there are no longer finitely many analytic types.In [Bri77], Brian¸con proves that Hilb d ( C [[ x, y ]]) is a ( d − x, y d + c d − y d − + · · · + c y ), with c , . . . , c d − ∈ k , form an open dense subsetof the punctual Hilbert scheme of k [[ x, y ]] of degree d . Such ideals correspond tolocal Artinian stretched k -algebras of multiplicity d , namely rings with maximal socle degree and hence Hilbert function { , , , . . . , } , see [EV08]. However, thereare other possible Hilbert functions for local rings of the same multiplicity. Weare interested in a description of points in Hilb d ( k [[ x, y ]]) that allows us to track alladmissible Hilbert functions and configurations of Betti numbers.Rossi and Sharifan prove in [RS10] that a minimal free resolution of k [[ x , . . . , x n ]] /J can be obtained from a sequence of zero and negative consecutivecancellations on the minimal free graded resolution of k [ x , . . . , x n ] / Lt τ ( J ). For n = 2, given a resolution of the lex-segment ideal Lex( h ) associated to a given Hilbertfunction h , they provide a procedure to explicitly realize ideals with any admissiblesequence of cancellations. This is done by considering very specific deformations ofa Hilbert-Burch matrix of Lex( h ). Our aim is to study all such deformations.In [CV08], Conca and Valla parametrize all ideals I in the polynomial ring P = k [ x, y ] that share the same leading term ideal with respect to the lexicographical orderby defining a canonical Hilbert Burch matrix of I . In [Con11], Constantinescu analo-gously parametrizes such sets of ideals for the degree lexicographical order wheneverthe leading term ideal is lex-segment. The fact that V τ ( E ) = { I ∈ P : Lt τ ( I ) = E } are affine spaces comes from a result by Bia lynicki-Birula in [Bia73]. If we consider V τ ( E ) for all monomial ideals E such that dim k P/E = d we obtain a decomposi-tion of Hilb d ( A k ). By analogy to Schubert cells in Grassmanians, the affine varieties V τ ( E ) are called Gr¨obner cells.This paper is devoted to the extension of this results to the local setting for thelocal order τ induced by the lexicographical order. Although local orders are nolonger well-orderings, there are analogous notions to Gr¨obner bases and Buchbergeralgorithm in the ring of formal power series and localizations of polynomial rings:standard bases and the tangent cone algorithm, see [Hir64],[Mor82], [GP08].Our main result, Theorem 4.7, gives a parametrization of all ideals in the powerseries ring R = k [[ x, y ]] with a special class of leading term ideals in terms of theirHilbert-Burch matrices. This class includes lex-segment ideals Lex( h ). Therefore, ifchar( k ) = 0, the parametrization of Lex( h ) already describes all ideals with Hilbertfunction h up to a generic change of coordinates, see Corollary 4.10.In this way, we generalize the procedure in [RS10] so that we can obtain all idealswith any given admissible number of generators arising from a deformation of amonomial ideal E . The Gr¨obner cells V ( E ) defined in this paper are compatiblewith the local structure. In other words, ideals with different Hilbert functions willnot be in the same cell, as opposed to the situation in Conca-Valla’s cells for m -primary ideals, see Example 4.11. In Conjecture 4.14 we suggest what should be the set of matrices giving aparametrization in the general case. An interesting application of a full parametriza-tion is the computation of all Gorenstein rings that are at a minimal distance of agiven Artin ring, see [EHM20].The first section of the paper reviews the existing results in P = k [ x, y ]. Section2 provides basic tools to transition from the polynomial case to the power seriesring. The third section is devoted to narrow down the set of candidates for canon-ical Hilbert-Burch matrices of the ideals in V ( E ). Combining Proposition 3.3 andProposition 3.9, we provide a surjection from the set of matrices N ≤ s ( E ) to the affinevariety V ( E ). The main result of the paper, Theorem 4.7 in Section 4, gives a bijec-tion between the set of matrices M ( E ) and the variety V ( E ) for special monomialideals E . We end the section with the conjecture for the general case. Plenty ofexamples are given to illustrate the behavior. Finally, the last section applies theparametrization to the computation of minimal Gorenstein covers of a given ring.1. Parametrization of ideals in k [ x, y ]Let k be an arbitrary field. Consider a monomial zero-dimensional ideal E in thepolynomial ring P = k [ x, y ]. By taking the smallest integer t such that x t ∈ E andthe smallest integers m i such that x t − i y m i ∈ E for any 1 ≤ i ≤ t , we can alwaysexpress such a monomial ideal as E = ( x t , x t − y m , . . . , x t − i y m i , . . . , y m t ) , where 0 = m < m ≤ · · · ≤ m t is an increasing sequence. If all the inequalities arestrict, we call E a lex-segment ideal.After fixing a term order, we can ask for all ideals I in P with leading termideal E . Reduced Gr¨obner bases provide a parametrization of this set of ideals.However, explicitly describing such a parametrization is not always straightforward.In [CV08], Conca and Valla consider a different approach. Instead of focusing onthe generators of I , they study the relations or syzygies among the generators. AHilbert-Burch matrix of the ideal I encodes these relations. Therefore, giving such aparametrization is equivalent to choosing a simple or canonical Hilbert-Burch matrixfor each ideal I . Definition 1.1.
The canonical Hilbert-Burch matrix of the monomial ideal E = ( x t , . . . , x t − i y m i , . . . , y m t ) is the Hilbert-Burch matrix of E of the form ROSER HOMS AND ANNA-LENA WINZ H = y d · · · − x y d · · · − x · · · ... ... ... · · · y d t · · · − x , where d i = m i − m i − for any ≤ i ≤ t . The degree matrix U of E is the ( t + 1) × t matrix with integer entries u i,j = m j − m i − + i − j , for ≤ i ≤ t + 1 and ≤ j ≤ t . It follows from the definition that u i,i = d i and u i +1 ,i = 1, for 1 ≤ i ≤ t .Conca-Valla parametrize the set V ( E ) of all zero-dimensional ideals I in P thatshare the same leading term ideal E with respect to the lexicographical term ordering.They give a set of matrices that deform the canonical Hilbert-Burch matrix of themonomial ideal E into Hilbert-Burch matrices of each I . We use the same notationas in [CV08]. Definition 1.2.
We denote by T ( E ) the set of matrices N = ( n i,j ) of size ( t + 1) × t with entries in k [ y ] such that • n i,j = 0 for any i < j , • deg( n i,j ) < d j for any i ≥ j . Theorem 1.3. [CV08, Theorem 3.3, Corollary 3.1]
Given a zero-dimensional mono-mial ideal E in P = k [ x, y ] with canonical Hilbert-Burch matrix H , the map Φ : T ( E ) −→ V ( E ) N I t ( H + N ) is a bijection. This theorem allows us to define the canonical Hilbert-Burch matrix of any zero-dimensional ideal I of P as H + Φ − ( I ), where H is the canonical Hilbert-Burchmatrix of the monomial ideal Lt lex ( I ).In [Con11], Constantinescu parametrizes the variety V deglex ( E ) = { I ⊂ P : Lt deglex ( I ) = E } , where the leading term ideals are considered with respect to the degree-lexicographical order, for E lex-segment. Definition 1.4.
Denote by A ( E ) the set of ( t +1) × t matrices A = ( a i,j ) with entriesin k [ y ] such that all its entries satisfy deg( a i,j ) ≤ ( min( u i,j + 1 , d i − , i ≤ j ;min( u i,j , d j − , i > j ; and u i,j are the entries of the degree matrix U of E . Theorem 1.5. [Con11, Theorem 3.1]
Given a zero-dimensional lex-segment ideal L in P = k [ x, y ] with canonical Hilbert-Burch matrix H , the map Φ : A ( L ) −→ V deglex ( L ) A I t ( A + H ) is a bijection. The proofs of well-definedness and surjectivity of Φ hold for any monomial ideal.Although the lex-segment hypothesis is needed in his proof of injectivity, the authorconjectures that Φ is a proper parametrization in the general case.2.
From polynomials to power series
We are now interested in the local setting. We want to parametrize zero-dimensional ideals of the ring of formal power series R = k [[ x, y ]] analogously tothe results of Conca-Valla and Constantinescu for the polynomial ring.One can still describe zero-dimensional monomial ideals of R as E =( x t , x t − y m , . . . , x t − i y m i , . . . , y m t ) and consider their canonical Hilbert-Burch matrix H as introduced in Definition 1.1. However, in order to define the leading term idealwe need to deal with term orders that are well-defined in a power series ring.2.1. Enhanced standard basis and Grauert’s division.Definition 2.1.
A term ordering τ in the polynomial ring P = k [ x , . . . , x n ] inducesa reverse-degree ordering τ in R = k [[ x , . . . , x n ]] such that for any monomials m, m ′ in R , m > τ m ′ if and only if deg( m ) < deg( m ′ ) or deg( m ) = deg( m ′ ) and m > τ m . We call τ the local degree ordering induced by the global ordering τ . Note that the local orders induced by the lexicographic and the degree lexico-graphic orders are the same. Moreover, in two variables, the latter also coincideswith the reverse degree lexicographic order.
ROSER HOMS AND ANNA-LENA WINZ
Definition 2.2.
Given an ideal J of R , we define the leading term ideal of J as the monomial ideal in P generated by the leading terms with respect to the localdegree ordering τ , i.e. Lt τ ( J ) = (Lt τ ( f ) : f ∈ J ) ⊂ k [ x, y ] . We call a subset { f , . . . , f m } of J a τ -enhanced standard basis of J if Lt τ ( J ) = (Lt τ ( f ) , . . . , Lt τ ( f m )) . Remark 2.3.
The term standard basis was first used by Hironaka in [Hir64, Def-inition 3] to refer to systems of generators of an ideal J in R whose initial formsgenerate the homogeneous ideal J ∗ . However, this terminology is not consistent inliterature and in other sources standard basis refer to what we here define as τ -enhanced standard basis, e.g. [GP08]. The notation used in this paper is the sameas in [Ber09]. Example 2.4.
Comparison between leading terms w.r.t. global and local orders.
Consider the lex-segment ideal L = ( x , x y, xy , y ) and set τ = lex. Let H be itscanonical Hilbert-Burch matrix and U its degree matrix: H = y − x y − x y − x , U = − . Consider the matrix M = H + N , where N is a 4 × , I = I ( M ) ⊂ P is an ideal in V ( L ). Indeed, the maximal minors of M give a τ -Gr¨obner basis { x − x , x y − xy, xy − y , y } of I and Lt lex ( I ) = L .However, the 3 × M are not a τ -enhanced standard basis of the ideal J = IR , namely the extension of I in the power series ring. In fact, J = ( x , xy, y )is itself a lex-segment ideal. The reason why the leading term ideal changes whencomputed with respect to τ is that n , = 1 has a term of degree lower than u , = 1. Remark 2.5.
Because Lt τ ( J ) is a monomial ideal, R/ Lt τ ( J ) has a natural gradedstructure. Moreover, we have R/J ≃ P/ ( J ∩ P ) for any zero-dimensional ideal J ⊂ R .Therefore we can consider Lt τ ( J ) indistinctively in R or P . The Hilbert function ofthe local ring R/J is defined asHF
R/J ( i ) = dim k (cid:18) ( m/J ) i ( m/J ) i +1 (cid:19) . Recall that HF
R/J = HF P/ Lt τ ( J ) , where the latter is the usual Hilbert function of agraded ring. On the contrary, when taking I ⊂ P and J ⊂ R from Example 2.4,note that HF R/J = { , , } whereas the Hilbert function of P/I is not even defined.Buchberger division can be replaced in the power series ring by Grauert’s division,see [Gra72]. Later on, Mora gave an analogous method to Buchberger’s algorithmin the local case: the tangent cone algorithm, see [Mor82]. We reproduce nexta modern formulation of Grauert’s division theorem in k [[ x , . . . , x n ]] from [GP08,Theorem 6.4.1]: Theorem 2.6. [Grauert’s Division Theorem] Let f, f , . . . , f t be in R . Then thereexist q , . . . , q t , r ∈ R such that f = t X i =0 q i f i + r satisfying the following properties: (1) No monomial of r is divisible by any Lt τ ( f i ) , for ≤ i ≤ t . (2) If q i = 0 , Lt τ ( q i f i ) ≤ τ Lt τ ( f ) . These techniques can be used to extend results that are well-understood for gradedalgebras to the local case. In [ERV14], they have been successfully applied to char-acterize the Hilbert function of one dimensional quadratic complete intersections.2.2.
Lifting of syzygies in local rings.
The connection between the lifting ofsyzygies and Gr¨obner bases has been widely studied in polynomial rings, see [KR00,Theorem 2.4.1]. Analogous results hold for rings of formal power series.Let F be a subset { f , . . . , f t } of R and set Lt τ ( F ) = { Lt τ ( f ) , . . . , Lt τ ( f t ) } . Bya slight abuse of notation, F and Lt τ ( F ) will be regarded as ( t + 1)-tuples of R t +1 when convenient. Mora, Pfister and Traverso prove in [MPT89, Theorem 3] that F is a τ -enhanced standard basis of an ideal of R if and only if any homogeneoussyzygy of Lt τ ( F ) can be lifted to a syzygy of F .For the sake of completeness, we will now give a precise definition of lifting in thissetting following the notation of [Ber09, Definition 1.7]. We define the degree of m = ( m , . . . , m t +1 ) ∈ R t +1 with respect to the ( t + 1)-tuple F ∈ R t +1 and the localordering τ asdeg ( τ , F ) ( m ) = max τ { Lt τ ( m i f i − ) : 1 ≤ i ≤ t + 1 and m i = 0 } . An element σ = { σ , . . . , σ t +1 } ∈ R t +1 is homogeneous with respect to ( τ , F )-degree if all its non-zero components reach the maximum leading term, namelyLt τ ( σ i f i − ) = deg ( τ, F ) ( σ ) for any i ∈ { , . . . , t + 1 } such that σ i = 0. ROSER HOMS AND ANNA-LENA WINZ
Definition 2.7.
We call m ∈ R t +1 a ( τ , F ) -lifting of a ( τ , F ) -homogeneous element σ ∈ R t +1 if m = σ + n , where n = ( n , . . . , n t +1 ) ∈ R t +1 satisfies (1) Lt τ ( n i f i − ) < τ deg ( τ, F ) ( σ ) for any ≤ i ≤ t + 1 such that n i = 0 . Conversely, we call σ the ( τ , F ) -leadingform of m and denote it by LF ( τ, F ) ( m ) = σ ∈ R t +1 . If both τ and F are clear from the context, we will just say that m is a lifting of σ , which in its turn is the leading form of m . The shift on the indexes of n and F in(1) is convenient for our specific setting, as we will see in the following example. Example 2.8.
Liftings of homogeneous elements.
Consider a monomial ideal E =( x t , x t − y m , . . . , y m t ) and take F = ( f , . . . , f t ) ∈ R t +1 such that Lt τ ( f i ) = x t − i y m i for any 0 ≤ i ≤ t . The columns σ , . . . , σ t of the canonical Hilbert-Burch matrix H of E are ( τ , F )-homogeneous elements with deg ( τ , F ) ( σ j ) = x t − j +1 y m j for any 1 ≤ j ≤ t .We can build liftings m j of σ j by taking m j = σ j + n j , where n j = ( n ,j , . . . , n t +1 ,j ) isa ( t + 1)-tuple of R t +1 such that either n i,j = 0 or Lt τ ( n i,j ) x t − i +1 y m i − < τ x t − j +1 y m j .As in the polynomial case, Bertella proves in [Ber09, Theorem 1.10] that themodule of syzygies of F is generated by liftings of homogeneous generators of themodule of syzygies of Lt τ ( F ). Recall that the fact that syzygies lift is equivalent tothe existence of a flat family I t where I = Lt τ ( F ) and I = ( F ), see [Ste03, Chapter1] and [MS05, Lemma 18.8].In the same paper, Bertella provides a very explicit characterization of τ -enhancedstandard bases in codimension two in terms of matrices that encode leading formsof the generators of the module of syzygies of the ideal: Theorem 2.9. [Ber09, Theorem 1.11]
Let M be a ( t + 1) × t matrix with entries in R . For ≤ i ≤ t , let f i be the determinant of M after removing row i + 1 and set F = ( f , . . . , f t ) . Let H be the matrix whose columns are the ( τ , F ) -leading forms ofthe columns of M . Assume that: • ht( f , . . . , f t ) = 2 , • I t ( H ) = (Lt τ ( f ) , . . . , Lt τ ( f t )) .Then the following are equivalent: (i) { f , . . . , f t } is a τ -enhanced standard basis of the ideal I t ( M ) . (ii) ht(Lt τ ( f ) , . . . , Lt τ ( f t )) = 2 . In other words, for zero-dimensional ideals J in R = k [[ x, y ]], a τ -enhanced standardbasis F arises from maximal minors of a Hilbert-Burch matrix M that encodes liftingsof syzygies of Lt τ ( F ). Towards a parametrization of ideals in k [[ x, y ]]From now on we will consider τ = lex. Definition 3.1.
Given a zero-dimensional monomial E ideal in R , we denote by V ( E ) the set of ideals J ⊂ R such that Lt τ ( J ) = E . Let us start by defining a set of matrices whose maximal minors generate all theideals with the same leading term ideal with respect to the local order τ . Definition 3.2.
Let E be a monomial ideal with canonical Hilbert-Burch matrix H and associated degree matrix U = ( u i,j ) . We define the set N ( E ) of ( t + 1) × t matrices N = ( n i,j ) with entries in k [[ y ]] such that all its non-zero entries satisfy ord( n i,j ) ≥ ( u i,j + 1 , i ≤ j ; u i,j , i > j. . Proposition 3.3.
Given a monomial ideal E = ( x t , . . . , x t − i y m i , . . . , y m t ) in R withcanonical Hilbert-Burch matrix H and degree matrix U , let V ( E ) be the set of idealsin Definition 3.1 and let N ( E ) be the set of matrices in Definition 3.2. The map ϕ : N ( E ) −→ V ( E ) N I t ( H + N ) is surjective. We prove this proposition in two steps: well-definedness in Lemma 3.4 and surjec-tivity in Lemma 3.5.
Lemma 3.4.
The map ϕ is well-defined.Proof. We need to prove that the leading term ideal Lt τ ( I t ( H + N )) is the monono-mial ideal E for any matrix N = ( n i,j ) in the set N ( E ).Consider the matrix M = H + N . The order bounds on the entries of N yieldord( m i,j ) ≥ (cid:26) u i,j + 1 , i < j ; u i,j , i ≥ j. Set f i = det[ M ] i +1 , for any 0 ≤ i ≤ t , where [ M ] i +1 is the square matrix that weget after removing row i + 1 of M . Since f i = X σ ∈ S t sgn( σ ) Y ≤ k ≤ t +1 , k = i +1 m k,σ ( k ) , we study the leading terms of polynomials of the form h = Q ≤ k ≤ t +1 , k = i +1 m k,σ ( k ) . If h is the product of all elements in the main diagonal of [ M ] i +1 , then Lt τ ( h ) = x t − i y m i . We claim that any other h = 0 satisfies Lt τ ( h ) < τ x t − i y m i . Indeed, sinceLt τ ( h ) = Y ≤ k ≤ t +1 , k = i +1 Lt τ ( m k,σ ( k ) ) , then ord( h ) = X ≤ k ≤ t +1 , k = i +1 ord( m k,σ ( k ) ) ≥ X ≤ k ≤ t +1 , k = i +1 u k,σ ( k ) . Equality can only be reached if subindices ( i, j ) satisfy i ≥ j , namely h = i Y k =1 ( y d k + n k,k ) t +1 Y k = i +1 m k,σ ( k ) , hence the maximal power of x is only reached at the main diagonal. Thus, any h = 0 away from the main diagonal satisfies Lt τ ( h ) < τ x t − i y m i and, therefore,Lt τ ( f i ) = x t − i y m i .Now we need to show that { f , . . . , f t } forms a τ -enhanced standard basis of I t ( M ).From the order bounds on the entries n i,j of N , it follows that the columns of M areliftings of the columns of H . See Example 2.8 for more details. By Theorem 2.9,it is enough to show that ht ((Lt τ ( f ) , . . . , Lt τ ( f t ))) = 2, which is clear because thisideal contains pure powers x t and y m t . Therefore, Lt τ ( I t ( M )) = E . (cid:3) Lemma 3.5.
The map ϕ is surjective.Proof. Consider a τ -enhanced standard basis { f , . . . , f t } of J ∈ V ( E ) such thatLt τ ( f i ) = x t − i y m i . We can assume that the monomials in the support of the f i ’s arenot divisible by x t , except for Lt τ ( f ).For any 1 ≤ j ≤ t , consider the S -polynomials S j := S ( f j − , f j ) = y d j f j − − xf j .Note that no monomial in Supp( S j ) is divisible by x t +1 for any 1 ≤ j ≤ t . ByTheorem 2.6 we have S j = t X i =0 q i,j f i , for some q i,j ∈ k [[ x, y ]] such that Lt τ ( q i,j f i ) ≤ Lt τ ( S j ). We claim that q i,j ∈ k [[ y ]].In fact, we will prove that this holds for any f ∈ J such that x t +1 does notdivide any monomial in Supp( f ). Assume LC τ ( f ) = 1. Consider such an f , thenLt τ ( f ) = x s y r for some 0 ≤ s ≤ t . On the other hand, from the fact that Lt τ ( f )belongs to Lt τ ( J ), it follows that x t − i y m i must divide Lt τ ( f ) for some 0 ≤ i ≤ t .Then t − i ≤ s and m i ≤ r , hence m t − s ≤ m i ≤ r . Define g = f − y r − m t − s f t − s . The new element g still belongs to J and satisfies again that none of its monomialsis divisible by x t +1 . In this way we can define a sequence ( g i ) i ∈ N , starting by g = f ,whose elements have decreasing leading terms with respect to τ . As in the proof ofGrauert’s division theorem in [GP08, Theorem 6.4.1], P i ∈ N g k converges with respectto the m -adic topology and f = X k ∈ N ( g k − g k +1 ) = t X i =0 X k ∈ N ,s k = t − i y r k − m t − sk ! f i . Therefore, for any 1 ≤ j ≤ t , the S -polynomial S j provides a relation betweengenerators of J y d j f j − − xf j + t +1 X i =1 n i,j f i − = 0 , where n i,j = − q i − ,j ∈ k [[ y ]]. This expression can be encoded in the matrix M = H + N , where N = ( n i,j ). From Lt τ ( n i,j f i − ) ≤ τ Lt τ ( S j ) it follows that any column m i of M is a lifting of a column σ i of H . The columns σ , . . . , σ t of H constitute ahomogeneous system of generators of Syz(Lt τ ( J )). Then, by [Ber09, Theorem 1.10], m , . . . , m t generate Syz( J ). The Hilbert-Burch theorem ensures that J is generatedby the maximal minors of M .Finally, the order bounds on the entries of N are obtained again fromLt τ ( n i,j f i − ) ≤ τ Lt τ ( S j ). Indeed, x t − i +1 y m i − + β i,j < τ x t − j +1 y m j , where Lt τ ( n i,j ) = y β i,j . Since(2) β i,j + t − i + 1 + m i − ≥ t − j + 1 + m j , we have β i,j ≥ i − j + m j − m i − = u i,j . If β i,j = u i,j , then equality holds in (2) andhence t − i + 1 < t − j + 1. In other words, β i,j ≥ u i,j and equality is only reachablewhen i > j . (cid:3) The proof of Lemma 3.5 provides a constructive method to obtain a matrix N ∈N ( E ) from any τ -enhanced standard basis { f , f , . . . , f t } of J ∈ V ( E ) such thatLt τ ( f i ) = x t − i y m i and x t does not divide any term of any f i except for Lt τ ( f ). Example 3.6.
Matrices in N ( E ) with proper power series entries. Set J = ( x + x y, y + x + x y ) and consider the τ -enhanced standard basis f = x + x y,f = x y + y ,f = x y ,f = xy ,f = y + x + x y. It can be checked that it satisfies the conditions of Lemma 3.5. The first S -polynomial is y f − xf = (cid:0)P i ≥ y i (cid:1) f + (cid:0)P i ≥ y i (cid:1) f − y f − (cid:0)P i ≥ y i (cid:1) f , hencesome entries in N are proper power series, not polynomials.Next we will see that, for any ideal J ∈ V ( E ), we can always find a matrix N ∈ N ( E ) with polynomial entries such that ϕ ( N ) = J . Example 3.7.
Obtaining matrices in N ( E ) with polynomial entries. The matrix in N ( E ) obtained from the τ -enhanced standard basis of J = ( x + x y, y + x + x y )given in Example 3.6 is N = − P i ≥ y i P i ≥ y i − P i ≥ y i P i ≥ y i y P i ≥ y i − P i ≥ y i . By removing all the terms of degree larger than 3 we get the matrix N = − y − y − y y + y + y − y y + y y − y with polynomial entries. Check that J = ϕ ( N ) = ϕ ( N ). Observe that, althoughthe behaviour with respect to the syzygies is much better, the τ -enhanced standardbasis of J given by the minors of H + N is less simple, for example ¯ f = x + x y + y − xy + y − x y − xy + y − x y − xy .For a general J ∈ V ( E ), we can only ensure that we will obtain the same ideal ifwe remove the terms in the entries of N with degree strictly higher than the socledegree of R/J , namely the largest integer s such that m s +1 ⊂ J . Definition 3.8.
Let E be a monomial ideal and let s be the socle degree of R/E .We define the set of matrices N ( E ) ≤ s := N ( E ) ∩ ( k [[ y ]] ≤ s ) ( t +1) × t . Proposition 3.9.
The restriction of ϕ to N ( E ) ≤ s is surjective.Proof. Consider J ∈ V ( E ), by Lemma 3.5 we know that J = I t ( H + N ) for some N ∈ N ( E ). Recall that J has the same Hilbert function as E , hence the socledegree of J is also s . We express N as N = N + e N , where N ∈ N ( E ) ≤ s and e N ∈ ( k [[ y ]] ≥ s +1 ) ( t +1) × t . We decompose e N into matrices e N i,j with at most one non-zero entry at position ( i, j ) such that e N = P i =1 ,...t +1 ,j =1 ,...,t e N i,j .By definition, J = ( f , . . . , f t ), where f k = det([ H + N ] k +1 ). Our goal is to provethat J = ( ¯ f , . . . , ¯ f t ), where ¯ f k = det([ H + N ] k +1 ).Let us use the Laplacian rule to rewrite the determinant. We denote by [ M ] ( l,m ) ,n the (square) submatrix of M that is obtained by deleting the l -th and m -th rows andthe n -th column. Then f k = det (cid:18)h H + N + P i,j e N i,j i k +1 (cid:19) = det (cid:16)(cid:2) H + N (cid:3) k +1 (cid:17) + P i,j ± ˜ n i,j · det (cid:16)(cid:2) H + N (cid:3) ( k +1 ,i ) ,j (cid:17) = ¯ f k + P i,j ± ˜ n i,j · det (cid:16)(cid:2) H + N (cid:3) ( k +1 ,i ) ,j (cid:17) . Since ˜ n i,j ∈ k [[ y ]] ≥ s +1 , it is clear that f i − ¯ f i ∈ ( x, y ) s +1 ⊂ J . Then J ′ = (cid:0) ¯ f , . . . , ¯ f t (cid:1) ⊂ J and, because Lt τ ( J ′ ) = Lt τ ( J ), we deduce that J = (cid:0) ¯ f , . . . , ¯ f t (cid:1) . (cid:3) It is important to note that Proposition 3.9 does not provide a parametrization of V ( E ). In general, the map ϕ is not injective even when we restrict it to N ( E ) ≤ s . Example 3.10.
The restriction of ϕ is not injective. Continuing Example 3.6 andExample 3.7, note that N ∈ N ≤ ( E ) but also N ′ = − y ∈ N ≤ ( E ) , with ϕ ( N ′ ) = ϕ ( N ) = J .The corresponding associated τ -enhanced standard basis of J is { x + x y, x y , x y , xy , y + x + x y } . Remark 3.11.
We have seen that ϕ : N ( E ) → V ( E ) as well as its restriction ϕ : N ≤ s ( E ) → V ( E ) are not injective. Although an ideal J can be obtained from dif-ferent matrices of the form H + N , the systems of polynomial generators { f , . . . , f t } of J that arise as maximal minors of any such matrices are all different. In otherwords, the map N ( E ) → R t +1 , that sends N to the maximal minors of H + N , isinjective.Indeed, if two matrices N, N ′ ∈ N ( E ) satisfy that the maximal minors of H + N and H + N ′ coincide, it follows that N = N ′ . The argument is the same as in thefirst paragraph of [Con11, 3.2] and we reproduce it here. Let { f , . . . , f t } be the maximal minors of H + N and H + N ′ . The columns of both matrices are syzygiesof { f , . . . , f t } , thence the columns of their difference H + N − ( H + N ′ ) = N − N ′ ∈ k [[ y ]] ( t +1) × t ) are also syzygies, but since the leading terms of the f i involve differentpowers of x , it follows that N = N ′ .4. Parametrization for lex-segment leading term ideals
A special situation occurs when a τ -enhanced standard basis of J and a lex-Gr¨obner basis of I = J ∩ P coincide. In this setting, we can overcome the lack ofinjectivity of ϕ : N ( E ) → V ( E ) by using Conca-Valla’s parametrization of V ( E ). Proposition 4.1.
Let J ∈ V ( E ) be an ideal that admits a τ -enhanced standard basis { f , . . . , f t } that is also a lex -Gr¨obner basis of I = J ∩ P with Lt τ ( f i ) = Lt lex ( f i ) .Then there exists a unique matrix N ∈ N ( E ) ∩ T ( E ) such that J = I t ( H + N ) .Proof. Let { f , . . . , f t } be a τ -enhanced standard basis of J that is also a lex-Gr¨obnerbasis with Lt τ ( f i ) = Lt lex ( f i ) = x t − i y m i . Then the f i are the signed maximal mi-nors of H + N for some N ∈ N ( E ) that is a strictly lower triangular matrix withpolynomial entries. Here by strictly lower triangular, we mean that n i,j = 0 for all i ≤ j .Assume that N is not yet in T ( E ), namely there exist ( i, j ) with deg( n i,j ) ≥ d j .In that case we decompose n i,j = r i,j + y d j q i,j with • u i,j ≤ ord( r i,j ) ≤ deg( r i,j ) ≤ d j − • max( u i,j − d j , ≤ ord( q i,j ) ≤ deg( q i,j ) ≤ deg n i,j − d j .Next we will perform the ( i, j )-reduction move defined in [Con11, Proof of 3] on N . Note that since N is strictly lower triangular, it corresponds to the second typeof reduction moves:Step 1. Add the j -th row multiplied by − q i,j to the i -th row of H + N .Step 2. Add the ( i − q i,j to the ( j − J and produces a new matrix e N whose( i, j )-entry has degree strictly less than d j . Checking that it preserves the orderbounds on the entries is a technicality that follows from the order bounds on r i,j and q i,j . Thus the matrix e N we obtain will still be in N ( E ) and the maximal minors of H + e N will also form a τ -enhanced standard basis of J .After performing finitely many reduction steps from the last to the first column,we will obtain a matrix N ∈ T ( E ) ∩ N ( E ) with J = I t ( H + N ). By Theorem 1.3, N is unique. (cid:3) This result allows us to extend the definition of canonical Hilbert-Burch matrix toany ideal that has a τ -enhanced standard basis { f , . . . , f t } that satisfies Lt τ ( f i ) = Lt lex ( f i ) = x t − i y m i . Moreover, the proof of Proposition 4.1 gives an algorithm toconstruct the canonical matrix from the matrix that encodes the S-polynomials of { f , . . . , f t } via reduction moves. Definition 4.2.
Set M ( E ) := N ( E ) ∩ T ( E ) . Let J ∈ V ( E ) be an ideal that admitsa τ -enhanced standard basis which is also a lex -Gr¨obner basis of I = J ∩ P . Wedefine the canonical Hilbert-Burch matrix of J as H + N , where N is the uniquematrix in M ( E ) such that J = I t ( H + N ) . Remark 4.3.
In [CV08], Conca and Valla provide parametrizations of certain sub-sets of V ( E ). V ( E ) is the set of all ( x, y ) − primary ideals I such that Lt lex ( I ) = E and it is parametrized by the set of matrices T ( E ) (see [CV08, Definition 3.2] foran explicit description). It is not difficult to check that M ( E ) = N ( E ) ∩ T ( E ) = N ( E ) ∩ T ( E ). Example 4.4.
Canonical Hilbert-Burch matrix.
Consider J = ( x , xy − y , y ) and E = Lt τ ( J ) = ( x , x y , x y , x y , x y , xy , y ). Set f = x , f i = x t − i y for i = 1 , . . . , f = xy − y and f = y . Note that { f , . . . , f } is a τ -enhancedstandard basis of J with Lt lex ( f i ) = Lt τ ( f i ) = x t − i y m i . The matrix H + N associatedto { f , . . . , f } is the following: y − x − x − x − x − x − y y − − x + y . The matrix N ∈ N ( E ) is strictly lower triangular, but since deg( n , ) = 3 ≥ d = 0and deg( n , ) = 0 ≥ d = 0, we see that N / ∈ T ( E ). By performing the reductionmoves (6 ,
5) and (7 , H + N of J ,with N ∈ M ( E ): M = H + N = y − x − x − x − x − x y − x + y . There is a class of monomial ideals E such that any ideal with leading term ideal E is under the hypothesis of Proposition 4.1: Lemma 4.5.
Let E = ( x t , x t − y m , . . . , y m t ) be a monomial ideal such that (3) m j − j − ≤ m i − i for all j < i. Then the reduced τ -enhanced standard basis of J ∈ V ( E ) is a Gr¨obner basis of I = J ∩ P with respect to the lexicographic term order and Lt lex ( I ) = E .Proof. Let { f i } i ∈I with I ⊂ { , . . . , t } be the unique reduced τ -enhanced standardbasis of J with Lt τ ( f i ) = x t − i y m i . There are two steps in this proof:( i ) Lt lex ( f i ) = x t − i y m i for any i ∈ I .Let us suppose that Lt lex ( f i ) = x k y l = x t − i y m i . Since x t − i y m i ∈ Supp( f i ), then x k y l > lex x t − i y m i and hence there are two possible situations: Case I: k = t − i and l > m i . Lt lex ( f i ) = x t − i y l is in the support of tail τ ( f i ) but x t − i y l ∈ E , which contradicts the reducedness hypothesis on { f j } j ∈I . Case II: k > t − i . Then we can set k = t − j for some 0 < j < i . Since Lt lex ( f i ) = x t − j y l and Lt τ ( f i ) = x t − i y m i , then t − i + m i = deg( x t − i y m i ) ≤ deg( x t − j y l ) = t − j + l. If there is an equality on the degree, the local order is equal to the lex order, henceLt τ ( f i ) = x t − j y l and we reach a contradiction. Therefore, we have t − i + m i < t − j + l .If l ≥ m j , the argument of Case I holds. Thus, we obtain the following sequence ofstrict inequalities t − i + m i < t − j + l < t − j + m j . It is equivalent to m i − i + 1 ≤ l − j ≤ m j − j − m j − j − ≤ m i − i , which leads to a contradiction.( ii ) { f i } i ∈I is a Gr¨obner basis of I with respect to lex.Since { f i } i ∈I is a subset of I , E = (Lt lex ( f i )) i ∈I ⊂ Lt lex ( I ). We can check thatLt lex ( I ) = E by looking at the dimensions. From R/J ∼ = P/I , it follows thatdim k ( P/ Lt lex ( I )) = dim k ( P/I ) = dim k ( R/J ) = dim k ( P/ Lt τ ( J )) = dim k ( P/E )and hence the inclusion E ⊂ Lt lex ( I ) becomes an equality. (cid:3) Remark 4.6.
Since for lex-segment ideals the sequence ( m i − i ) i is strictly increasing,lex-segment ideals satisfy 4.5. But the class of ideals is bigger. For example idealswith equality m i = m i +1 for exactly one i satisfy this condition too. Theorem 4.7.
Let E = ( x t , . . . , x t − i y m i , . . . , y m t ) be a lex-segment ideal (or an idealsatisfying condition (3)). Let H be the canonical Hilbert-Burch matrix of E . Thenthe restriction of the map ϕ from Proposition 3.3 to M ( E ) ϕ : M ( E ) −→ V ( E ) N I t ( H + N ) is a bijection.Proof. The map ϕ is well-defined by Lemma 3.4. Lemma 4.5 and Proposition 4.1ensure the existence of a unique matrix N ∈ M ( E ) such that J = I t ( H + N ). (cid:3) Note that when E is an ideal satisfying (3), then the set M ( E ) has a simpledescription. It is formed by matrices of size ( t + 1) × t with entries in k [ y ] such that n i,j = (cid:26) , i ≤ j ; c v i,j i,j y v i,j + c v i,j +1 i,j y v i,j +1 + · · · + c d j − i,j y d j − , i > j ;where v i,j := max( u i,j , Corollary 4.8.
Let E be the lex-segment ideal ( x t , x t − y m , . . . , y m t ) (or an idealsatisfying condition (3)) with degree matrix U = ( u i,j ) , v i,j = max( u i,j , and d j = m j − m j − for any ≤ i ≤ t + 1 and ≤ j ≤ t . Then V ( E ) is an affine space ofdimension N , where N = X ≤ j +1 ≤ i ≤ t +1 ( d j − v i,j ) . Let us show the details of the parametrization of the Gr¨obner cell V ( E ) as anaffine space A Nk with an example: Example 4.9.
Gr¨obner cell of a lex-segment ideal.
Consider the lex-segment ideal L = ( x , x y, xy , y ). By Theorem 4.7, any canonical Hilbert-Burch matrix M = H + N , with N ∈ M ( L ), associated to an ideal J ∈ V ( L ) is of the form M = y − x y c , − x + c , y y c , c , + c , y − x + c , y . We identify any ideal J = I ( M ) with the point p J = ( c , , c , , c , , c , , c , , c , ) ∈ A k . In other words, V ( L ) can be identified with the affine space A k . Note that thepoint at the origin in A k corresponds to the monomial ideal L . Corollary 4.10.
Assume char( k ) = 0 and let h be an admissible Hilbert function.Let L = Lex( h ) be the unique lexicographical ideal such that HF R/L = h . Then anyideal J ⊂ R such that HF R/J = h is of the form I t ( H + N ) , for some N ∈ M ( L ) ,after a generic change of coordinates.Proof. It follows from Theorem 4.7 and the fact that for any J ⊂ R such thatHF R/J = h it holds Lex( h ) = Gin τ ( J ). Here Gin τ ( J ) is the extension to the localcase defined in [Ber09, TheoremDefinition 1.14] of the usual notion of generic initialideal. (cid:3) Example 4.11.
Two stratifications of
Hilb ( k [[ x, y ]]) . There are three monomialideals of colength 3 in two variables: E = ( x, y ), E = ( x , xy, y ) and E = ( x , y ).The punctual Hilbert scheme Hilb ( k [[ x, y ]]) can be stratified into three correspondingGr¨obner cells that depend on the term ordering that we choose. The following tabledescribes the ideals that we find in each Gr¨obner cell with respect to lex, namely V ( E i ), and the induced local order, namely V ( E i ), with i = 1 , ,
3. Recall that V ( E i ) is the affine space in Conca-Valla parametrization introduced in Remark 4.3that only considers m -primary ideals in the polynomial ring, hence it provides aproper stratification of Hilb ( k [[ x, y ]]). E i E = ( x, y ) E = ( x , xy, y ) E = ( x , y )HF R/E i { , , } { , } { , , } τ J = ( x, y + c y + c y ) J = ( x , xy, y ) J = ( x , y + cx ) τ = lex I = ( x, y + c y + c y ) I = ( x + cy, xy, y ) I = ( x , y )The extension J = IR of ideals I ∈ V ( E ) with c = 0 are of the form J =( y − c x , x ) ∈ V ( E ). Note that HF R/J = HF R/E , hence ideals in the same Gr¨obnercell with respect to the lexicographical order can have different Hilbert functionswhen considered in the power series ring. By construction this will never happenin Gr¨obner cells with respect to the local order. In this sense we say that theparametrization given in Theorem 4.7 is compatible with the local structure.In the general case, we have a surjective map ϕ : N ( E ) ≤ s → V ( E ). Restricting to M ( E ) we get an injection to V ( E ), but if E does not satisfy condition (3) the map ϕ is not surjective anymore. Lemma 4.12. If E does not satisfy condition (3), then there exists J ∈ V ( E ) suchthat Lt lex ( J ∩ P ) = E .Proof. Since condition (3) is not satisfied, there exist j < i such that(4) m j − j − > m i − i. Take i ′ = max { l | m i = m l } and j ′ = min { l | m j = m l } , then m j ′ − j ′ − > m j − j − > m i − i > m i ′ − i ′ . Replace i with i ′ and j with j ′ . Note that (4) still holds and now additionally d j ≥ d i +1 ≥ f k = x t − k y m k for k ∈ { , . . . , t }\{ i } and f i = x t − i y m i + x t − j y m j − . Consider theideal J = ( f , . . . , f t ) of R . Clearly, Lt lex ( f i ) = x t − j y m j − / ∈ E , thus Lt lex ( J ∩ P ) = E .Now we need to prove that Lt τ ( J ) = E . From (4) we have t − i + m i < t − j + m j − τ ( f i ) = x t − i y m i . The polynomial f i cannot be reduced by the other (monomial)generators.The S -polynomials are S l = − x t − j +1 y m j − , l = i ; x t − j y m j − d i +1 , l = i + 1;0 , otherwise.If i < t , check that S i = y d j − f j − and S i +1 = y d i +1 − f j . Then the matrix N hasonly two non-zero entries n j,i = y d j − and n j +1 ,i +1 = − y d i +1 − . If i = t there is onlyone non-zero S -polynomial. In any case, one can check that N ∈ N ( E ). Thence, { f , . . . , f t } forms a τ -enhanced standard basis and J ∈ V ( E ). (cid:3) Example 4.13. M ( E ) → V ( E ) not surjective. Consider E = ( x , xy , y ) as inExample 4.4. E does not satisfy condition (3) because for ( i, j ) = (5 ,
1) we have m − − > m − − −
3. The ideal J from Lemma 4.12 in this case isgenerated by the monomials x − k y m k for k = 0 , . . . , , xy + x y . J ∩ P / ∈ V ( E )because Lt lex ( J ∩ P ) = ( x , x y, x y , xy , y ). Therefore, J / ∈ ϕ ( M ( E )).Several computations, comparison to [Con11], considerations about the reductionmoves and a detailed study of complete intersections give us strong evidence of whatthe subset of N ( E ) ≤ s that provides a bijection should be.We define the subset ( k [ y ]
Let E be a monomial ideal. Then the set N ( E ) An approach analogous to Proposition 4.1 and [Con11, Proof of 3]with reduction moves does not work in general. It can be verified that if we startwith any N ∈ N ( E ) the matrix obtained by a reduction move is in N ( E ). If thematrix is additionally strictly upper or strictly lower triangular, there is an obviousorder in which one can perform the reduction moves to obtain a matrix in N ( E ) 10 0 0 − x . Applications to the construction of Gorenstein rings Let us assume that k is a field of characteristic 0. The explicit description ofthe affine variety V ( L ) given by Theorem 4.7 allows us to parametrize Gorensteinrings R/J with a given Hilbert function h up to a generic change of coordinates. Itis enough to consider those Gorenstein ideals J that arise as a deformation of theunique lex-segment ideal L = Lex( h ) associated to h . We will now see that the subset V G ( L ) of all Gorenstein ideals in V ( L ) has the structure of a quasi-affine variety. Proposition 5.1. Let L be a lex-segment ideal and let J be an ideal with Lt τ ( J ) = L . Let H and M = H + N be the canonical Hilbert-Burch matrices of L and J ,respectively. Then J is Gorenstein if and only if the third main diagonal of N consistsof polynomials in y with non-zero constant terms.Proof. In codimension 2, J is Gorenstein (equivalently, complete intersection) if andonly if it is minimally generated by 2 elements. Let M be the matrix whose entriesare the classes of the entries of M in R/ m . By [Ber09, Lemma 2.1], J is Gorensteinif and only if rk( M ) = t − 1. It can be checked easily that this is equivalent to c , c , · · · c t +1 ,t − = 0, where c i,i − is the constant term of the entry n i,i − of N . (cid:3) Remark 5.2. Proposition 5.1 provides a method of determining whether a lex-segment ideal L admits Gorenstein deformations by looking at the degree matrix U of the canonical Hilbert-Burch matrix H of L . Gorenstein ideals are admissible ifand only if u i,i − ≤ ≤ i ≤ t + 1. See [Ber09] for details on what theadmissible Hilbert functions for Gorenstein rings of codimension 2 are. Example 5.3. Parametrization of Gorenstein deformations of a lex-segment ideal. Consider L = ( x , x y, xy , y ). From Example 4.9 we have M = c , c , c , . By Proposition 5.1, J = I ( M ) is Gorenstein if and only if c , c , = 0. Then the setof Gorenstein ideals J with Lt τ ( J ) = L can be identified with A k \ V ( c , c , ). Corollary 5.4. Let L be a lex-segment ideal. The set V G ( L ) of Gorenstein ideals J such that Lt τ ( J ) = L is a quasi-affine variety. Remark 5.5. Corollary 5.4 is a generalization of the procedure given in [RS10,Remark 4.7] by Rossi and Sharifan to explicitly construct a Gorenstein ring J whoseresolution is obtained by consecutive and zero cancellation of the resolution of L =Lex( h ).The parametrization of Gorenstein ideals can also be used to find Gorenstein Artinrings G = R/J that are as close as possible to a given Artin ring A = R/I . See[Ana08],[EH18],[EHM20] for more details on this problem. Definition 5.6. We call the Artin Gorenstein ring G = R/J a minimal Goren-stein cover of the Artin ring A = R/I if J ⊂ I and dim k G − dim k A is minimalamong all Artin Gorenstein rings mapping onto A . The difference dim k G − dim k A is called the Gorenstein colength of A , denoted by gcl( A ) . Let us show through an example how we can find such Gorenstein covers usingthe canonical Hilbert-Burch matrices provided by Theorem 4.7: Example 5.7. Parametrization of minimal Gorenstein covers of A = R/I arisingfrom a lex-segment ideal. Consider the ideal I = ( x − xy , x y − y , y ) with Hilbertfunction { , , , } . The sequence h = { , , , , } corresponds to the Hilbert func-tion of smallest length that admits Gorenstein ideals J where the inclusion J ⊂ I is possible a priori. The lex-segment ideal associated to h is our running example L = ( x , x y, xy , y ), see Example 4.9 and Example 5.3.On one hand, the inclusion condition J ⊂ I can be described by a normal formcomputation of the generators of J ∈ V ( L ) ≃ A k with respect to a standard basis of I . The point p J = ( c , , c , , c , , c , , c , , c , ) ∈ A k satisfies the inclusion property ifand only if it belongs to the affine variety V ( − c , + c , c , − c , + 2 , c , + c , ) ⊆ A k .On the other hand, V G ( L ) ≃ A k \ V ( c , c , ). Therefore, J is a Gorenstein cover of A if and only if p J ∈ V ( − c , + c , c , − c , + 2 , c , + c , ) \ V ( c , c , ).For instance, the point (1 , , , , , ∈ A k corresponds to the Gorenstein cover G = R/ ( x y − y , x − xy ) of A . In particular, we proved that gcl( A ) = 2. Corollary 5.8. The set of Gorenstein covers of G = R/J of A = R/I that arisefrom a deformation of a lex-segment ideal L , namely Lt τ ( J ) = L , is a quasi-affinevariety. Remark 5.9. Not all minimal Gorenstein covers come from deformations of a lex-segment ideal. The reason behind this is that the inclusion condition J ⊂ I is notpreserved after a generic change of coordinates on J . To make sure we do not missany Gorenstein cover we need to look also at deformations of all monomial ideals with a convenient Hilbert function. The surjectivity of Proposition 3.9 is enough todetect the existence of minimal Gorenstein covers and hence enough to compute theGorenstein colength. Examples can be found in [Hom19]. However, to compute thequasi-projective variety of minimal Gorenstein covers defined in [EHM20, Theorem4.2], a proper parametrization for the general case would be desirable. Acknowledgements We want to thank Alexandru Constantinescu for suggesting the problem and forhelpful discussions, as well as for recommending to the second author a stay at theUniversit`a Degli Studi di Genova. The first author wants to thank Joan Elias forencouraging her to do a research stay with Maria Evelina Rossi. We also want tothank Bernd Sturmfels for his advice.We want to give special thanks to Maria Evelina Rossi for hosting us in Genova,answering many questions, giving useful hints and commenting on several versions ofthis manuscript, and especially for suggesting us to work on this problem together.The first author was partially supported by MTM2016-78881-P, BES-2014-069364and EEBB-I-18-12915. Travel of the second author to Genova was supported by aVigoni project. 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