Equigenerated Gorenstein ideals of codimension three
aa r X i v : . [ m a t h . A C ] J u l Equigenerated Gorenstein ideals of codimension three
Dayane Lira Zaqueu Ramos Aron Simis Abstract
One focus on the structure of a homogeneous Gorenstein ideal I of codimensionthree in a standard polynomial ring R = k [ x , . . . , x n ] over a field, assuming that I isgenerated in a fixed degree d . This degree is tied to the minimal number of generatorsof I and the degree of the entries of an associated skew-symmetric matrix and oneshows, conversely, in a characteristic-free way, that any such data are fulfilled by someGorenstein ideal. It is proved that a codimension 3 homogeneous Gorenstein ideal I ⊂ k [ x, y, z ] cannot be generated by general forms of fixed degree, except preciselywhen d = 2 or the ideal is a complete intersection. The question as to when the link( ℓ m , . . . , ℓ mn ) : f is equigenerated, where ℓ , . . . , ℓ n are independent linear forms and f isa form, is given a solution in some important cases. Introduction
Much has been said about Gorenstein ideals (for a tiny list, see [2], [29], [6], [11], [17], [18],[24], [25], [19], [22], [21], [9], [23], [20]) and yet there seems to be more to claim, even inthe thoroughly studied case where the codimension is 3. The objective of this work is tounderstand, and possibly reveal, a few new aspects of a codimension 3 Gorenstein ideal I ⊂ R generated by forms of same degree (equigenerated, as one says), where R := k [ x , . . . , x n ]is a polynomial ring over a field k .There is nothing definite not even about the first case ( n = 3) (ternary ideal, as onesays). It seems like a good shot starting there, in which case one is in the whelm of standardgraded Artinian Gorenstein rings over a field. Do we know nearly everything about theserings, or at least how it impacts knowing that they have a presentation by means of anequigenerated ideal?We have been inspired by the classification ideas of [21] and [15]. A fairly understoodcase is that of a presentation by means of an ideal I ⊂ R of finite colength generated byquadrics (see [15]). If R = k [ x, y, z ], the minimal number of generators µ ( I ) of I in this caselies on the range 3 ≤ µ ( I ) ≤
6. The extreme values are well-known (complete intersectionand m , respectively). The non-trivial cases of µ ( I ) = 4 , µ ( I ) = 5, it comes with a bit of surprise in thefollowing: Theorem. ([15]) Let k be a field of characteristic = 2 and let I ⊂ R = k [ x, y, z ] be an idealof finite colength generated by quadrics. Then I is a Gorenstein ideal if and only if it issyzygetic. AMS 2010 Mathematics Subject Classification (2010 Revision). Primary 13H10, 13D02; Secondary13A30, 13C13, 13C15, 13C40. Key Words and Phrases : Gorenstein ideal, socle degree, Macaulay inverse, Newton dual, space of pa-rameters, general forms. Under a PhD fellowship from CAPES, Brazil (88882.440720/2019-01) Under a post-doc fellowship from INCTMAT/Brazil (88887.373066/2019-00) Partially supported by a CNPq grant (302298/2014-2)
1t is a general fact that, over a regular ring where 2 is invertible, a codimension 3Gorenstein ring is syzygetic. Thus, the theorem is really about the inverse implication.Note that it says nothing about as to whether there are actually examples of Gorensteinideals in the case where µ ( I ) = 5 – or syzygetic ideals, for that matter. A prototype ofsuch an example is the ideal ( xy, xz, yz, x − y , x − z ). The reader may actually wonderwhich one theoretically comes out easier in this example: Gorenstein or syzygetic? In fact,all the others fit a suitable generic version of this one, according to: Proposition ([15, Theorem 3.6]) Let I ⊂ k [ x, y, z ] denote an m -primary ideal minimallygenerated by 5 quadrics. Then I is Gorenstein if and only if there exist 3 independentlinear forms { l , l , l } ⊂ k [ x, y, z ] such that I = ( xl , xl , yl , xl − yl , xl − zl ) . Of course, making explicit a minimal set of generators can only lead us so far, since onemay not easily recognize traces of the usual characterizations of a codimension 3 Gorensteinideal. One knows that a codimension 3 Gorenstein ideal generated by quadrics has asminimal set of generators the maximal Pfaffians of a 5 × I out of the above generators, but deriving a skew-symmetric one whosemaximal Pfaffians generate I is about of the same difficulty.A second aspect is that, although not immediately derived from the above proposition,being Gorenstein of codimension 3 and generated by quadrics is a Zariski dense open condi-tion in the space of parameters – this will be proved in Theorem 2.6 and is exclusive of thisdegree. Thus, the theorem shows that being Gorenstein of codimension 3 and generated by r ≥ d ≥ µ ( I ) of generators of a codimension 3 Gorenstein ideal I ⊂ k [ x, y, z ]generated in degree d obeys a certain additional constraint, besides being an odd integer.By drawing on a preliminary obstruction introduced in Section 3.1, one sees, e.g., that for d = 3, among the possibilities µ ( I ) = 5 , µ ( I ) = 7 is the effective alternativearound the corner.This obstruction is formalized in terms of a numerical virtual datum , to wit, a pair ( d, r )of integers such that d ≥ r ≥ equigenerated Gorenstein virtualdatum in dimension n ≥ r − / d . Such a datum will be said to be proper if there exists a codimension 3 Gorenstein ideal in k [ x , . . . , x n ] generated by r formsof degree d , in which case 2 d/ ( r −
1) is in fact the degree of any entry in the alternatingmatrix whose maximal Pfaffians generated the ideal. The obstruction nature of this notionwill soon be discarded throughout because of Proposition 3.11, which says that any virtualdatum is proper. This result is characteristic-free. It requires a kind of generic arguing inthe proof, but is surprisingly not in conflict with Theorem 2.6 previously mentioned. Thisquestion has been tackled before in [4] in a different way, based upon a particular skew–symmetric matrix. An approach of the same sort, drawing upon the celebrated example in[2, Proposition 6.2], will appear in the PhD thesis of the first author.A seemingly more difficult question in this context is whether, given a virtual datum2 d, r ), there are only finitely many codimension 3 Gorenstein ideals with this datum, up tocoordinate change in the k -vector space k x + · · · + k x n . Moreover, in the affirmative casecan one write down respective canonical forms? In addition, an answer to this questionmay depend on the nature of the base field k . But, even if affirmative for two given fields,the number of such ideals may differ from one another. These questions have been, nearlysimultaneously and independently, considered in [21] and [15], but it would look like thereis still much ground to be covered.The latter questions, along with the role of syzygetic ideals, will hopefully be tackledby the authors in a future paper. For the present, this role comes out in a certain behaviorof the algebras associated to a codimension 3 d -equigenerated Gorenstein ideal I . Therelation between the depth of these algebras and the nature of the rational map defined bythe linear space I d is discussed in Section 3 by completely elementary ways. There is noclaim of complete originality in the results themselves, with a strong chance that some becontained in some form in the work of Kustin–Polini–Ulrich through the encyclopedic [23].Another subject focused in this paper is what we call the ( x m , . . . , x mn )- colon problem ,for lack of better terminology and to avoid overusing ‘link’ instead. It has long been known(see [1, Proposition 1.3]) that, in arbitrary characteristic, any homogeneous Gorenstein idealof codimension 3 in k [ x , . . . , x n ] can be obtained as a colon ideal ( x m , . . . , x mn ) : f , for someinteger m ≥ f . In this regard, two questions naturally come up: first, is therea more definite relation between f and the ideal I ? Second, is there a characterization as towhen the resulting Gorenstein ideal is equigenerated in terms of the exponent m and theform f ?We give an answer to the first question, at least in characteristic zero, in terms ofMacaulay inverses and the notion of Newton duality as introduced in [5] and [7]. Thisis the content of Proposition 4.1 which says that Macaulay inverse to I is the (socle-like)Newton dual of the form f . The proposition also gives a complete characterization of f ,and its degree, under the condition that none of its nonzero terms belongs to the ideal( x m , . . . , x mn ).The second problem above is more delicate than what it surfaces. For a feeling, considerthe ideal ( x d , y d , z d ) : ( x + y + z ) d − which, for any d ≥ d ([21]); however, e.g., for d = 4, the ideal ( x d , y d , z d ) : x d − + y d − + z d − is not equigenerated as it contains the minimal generator xyz . Thus,merely controlling degrees in both terms of the colon operation is not enough.We solve this question in the case where I has linear resolution (Theorem 4.4), in termsof a degree constraint and show that, when the directrix f is a power of x + · · · + x n , thesolutions lie on a dense open set of the space of parameters (Proposition 4.5). The mainpart of the latter requires zero characteristic because it invokes a beautiful Lefschetz kindresult of R. Stanley, with algebraic proofs by Reid–Roberts–Roitman ([26] and Oesterl´e in([3, Apendix A])).In the way of considering the last problem, one introduces the notion of pure powergap , an integer measuring how far off from the socle degree is an exponent of the powers ofindependent linear forms lying in the ideal. We give the basic role of this invariant, hopingit will be useful in other contexts. Acknowledgment:
Upon posting on the arXiv a first version of this work, it hasbeen brought to our knowledge by A. Iarrobino that a couple of our results have been3onsidered before in [6] and [4]. We thank him very much for pointing our missing priorwork. Since our approach is often different, we decided to keep those results anyway, whilegiving appropriate priority.
Let R = k [ x , . . . , x n ] be a polynomial ring over a field k . Denote by m the maximalhomogeneous ideal of R. Given a homogeneous m -primary ideal I ⊂ R , consider the leastinteger s such that m s +1 ⊂ I. Then, the graded k -algebra R/I can be written as
R/I = k ⊕ ( R/I ) ⊕ ( R/I ) ⊕ · · · ⊕ ( R/I ) s with ( R/I ) s = 0 . The socle of R/I, denoted Soc(
R/I ) , is the ideal I : m /I ⊂ R/I . Since I and m are homogeneous ideals, then Soc( R/I ) is a homogeneous ideal of
R/I.
In particular,Soc(
R/I ) = Soc(
R/I ) ⊕ · · · ⊕ Soc(
R/I ) s . The integer s is the socle degree of R/I.
It is well known that
R/I is a Gorenstein ring ifand only if Soc(
R/I ) = Soc(
R/I ) s and dim k Soc(
R/I ) s = 1 . This degree is easily determined from the numerical data of the ideal (Lemma 3.2).
One briefly recalls the so–called Macaulay inverse system. It would be unearthly to give aspecific reference, given the amount that has been written about it. In any case, see [21],[8] and [9] for some excellent recent accounts.Let V be a vector space of dimension n over a field k and let x , . . . , x n be a basis for V. Let R = Sym k ( V ) = k [ x , . . . , x n ] be the standard graded polynomial ring in n variablesover k . Set y , . . . , y n for the dual basis on V ∗ = Hom k ( V, k ) and consider the divided powerring D k ( V ∗ ) = M i ≥ Hom( R i , k ) = k DV [ y , . . . , y n ] . In particular, { y [ α ] | α ∈ N n and | α | = j } is the dual basis to { x α | α ∈ N n and | α | = j } on D k ( V ∗ ) j = Hom( R j , k ) . If α ∈ Z n then one sets y [ α ] = 0 if some component of α is negative.Make D k ( V ∗ ) into a module over R through the following action R × D k ( V ∗ ) → D k ( V ∗ ) , ( f = X α a α x α , F = X β b β y β ) f F = X α,β a α b β y [ β − α ] . For a homogeneous ideal I ⊂ R and an R -submodule M ⊂ D k ( V ∗ ) one defines:Ann( I ) := { g ∈ D k ( V ∗ ) | Ig = 0 } and Ann( M ) := { f ∈ R | f M = 0 } . Then Ann( I ) is an R -submodule of D k ( V ∗ ), while Ann( M ) is an ideal of R. The R -module Ann( I ) is called the Macaulay inverse system of I. Theorem.
There exists a one-to-one correspondence between the set of non-zero ho-mogeneous height n Gorenstein ideals of R and the set of non-zero homogeneous cyclicsubmodules of D k ( V ∗ ) given by I Ann( I ) with inverse M Ann( M ) . Moreover, thesocle degree of
R/I is equal to the degree of a homogeneous generator of Ann( I ) . Let R = k [ x ] = k [ x , . . . , x n ] denote a polynomial ring over a field k . Fix an integer d ≥ { f , . . . , f r } ⊂ R be forms of degree d generating an ideal of codimension n . Considerthe parameter map R d × · · · × R d → P N × · · · × P N f = ( f , . . . , f r ) P f = ( λ (1) d, ,..., : · · · : λ (1)0 ,..., ,d ; . . . ; λ ( r ) d, ,..., : · · · : λ ( r )0 ,..., ,d ) (1)where N = dim k R d − (cid:0) d + n − d (cid:1) − f t = X | α | = d λ ( t ) α x α for each 1 ≤ t ≤ r. Given an integer e ≥ , denote by D the dimension of the k -vector space R d + e . The k -vector subspace R e f + · · · + R e f r of R d + e is spanned by the following set { x α f | | α | = e } ∪ · · · ∪ { x α f r | | α | = e } In particular, the coefficient matrix of this set with respect to canonical basis of R d + e is a D × r ( N + 1) matrix with the following shape: (cid:0) M . . . M r (cid:1) where M t is the coefficient matrix of the set { x α f t | | α | = e } with respect to canonical basis of R d + e . In particular, the entries block M t involve only the coefficients ( λ ( t ) d, ,..., , . . . , λ ( t )0 ,..., ,d )in the t th copy of P N .Let { Y ( t ) d, ,..., , . . . , Y ( t )0 ,..., ,d } denote the coordinates of the t th copy of P N and let M d,r,e stand for the matrix whose entries are the k -linear forms obtained by replacing each λ tα inthe block M t above by the corresponding Y tα . Then the generators of the ideal I D ( M d,r,e )of D -minors are multihomogeneous polynomials in the multigraded polynomial ring k [ Y (1) d, ,..., , . . . , Y (1)0 ,..., ,d ; . . . ; Y ( r ) d, ...,, , . . . , Y ( r )0 ,..., ,d ] . Proposition 2.1. R e f + · · · + R e f r ( R d + e if and only if P f is a point in the multiprojectivesubvariety V ( I D ( M d,r,e )) ⊂ P N × · · · × P N . Proof.
The dimension of the k -vector subspace R e f + · · · + R e f r is exactly the rank ofthe matrix M d,r,e evaluated in P f . ✷ .2 Examples of proper parametrization A major question is when V ( I D ( M d,r,e )) is a proper subvariety.Next are some examples of monomial ideals of finite colength in k [ x, y, z ], where thisquestion is affirmatively answered. The examples will in turn have a role in the subsequentsection. Proposition 2.2.
Consider the following ideal of R = k [ x, y, z ] I = ( z ( x, y ) d − , x ( x, y ) d − , y d , z d ) , where d ≥ . Then: (a) The minimal graded free resolution of
R/I is of the form → R ( − ( d + 2)) d − ⊕ R ( − ( d + 3)) ⊕ R ( − (2 d )) d − → R ( − ( d + 1)) d − ⊕ R ( − ( d + 2)) ⊕ R ( − (2 d − d → R ( − d ) d +1 → R (b) The decomposition structure of the socle of
R/I is as follows:
Soc(
R/I ) = (cid:26) k ( − ⊕ k ( − , if d = 2 k ( − ( d − d − ⊕ k ( − d ) ⊕ k ( − (2 d − d − , if d ≥ If d ≥ then V ( I α ( M d, d +1 ,d − )) is a proper subset of P N × · · · × P N . Proof. (a) For convenience, we write the generators of I in the following block shapedmatrix: φ := (cid:2) z f x g y d z d (cid:3) , where f := [ x d − x d − y · · · y d − ] and g := [ x d − x d − y · · · y d − ]. Then φ is the matrix ofthe map R ( − d ) d +1 → R .Throughout e j and i × j will denote the identity matrix of order j and the i × j nullmatrix, respectively. Consider further the following matrices, which will be candidates tofirst and second syzygies. φ = a a a a d × ( d − d × a b ( d − × b ( d − × b b ( d − × d × ( d − × × ( d − c × ( d − c × d × ( d − × × ( d − × × ( d − × d , with three blocks in standard degrees 1 , d − degree blocks .Here a = (cid:20) y e d − × ( d − (cid:21) , a = ( d − × − yx , a = (cid:20) x e d − × ( d − (cid:21) , a = (cid:20) ( d − × y (cid:21) , a = − z d − e d = (cid:20) × ( d − − z e d − (cid:21) , b = − z e d − , b = − y · · · x − y · · · ... . . . . . . ... · · · x − y · · · x , b = (cid:20) ( d − × − y (cid:21) c = (cid:2) − z (cid:3) , c = (cid:2) x (cid:3) , d = (cid:2) x d − x d − y · · · y d − (cid:3) . The second matrix is φ = s ( d − × u ( d − × × ( d − t × ( d − v s t u ( d − × × d − t × ( d − × s ( d − × ( d − × ( d − ( d − × × ( d − t × ( d − × d × ( d − d × u v , where s = x e d − , s = (cid:20) − y e d − × ( d − (cid:21) , s = z e d − , t = (cid:2) − xy (cid:3) , t = (cid:20) ( d − × − y (cid:21) , t = (cid:2) x (cid:3) , t = (cid:2) z (cid:3) , u = − z d − e d − , u = (cid:20) × ( d − z d − e d − (cid:21) , u = − y · · · x − y · · · ... . . . . . . ... · · · x − y · · · x . . . v = (cid:2) z d − (cid:3) , v = ( d − × − yx The goal of this seemingly bizarre block wise way of writing matrices is to adjust pagefitting.
Claim 1.
The sequence of R -maps0 → R d − φ −→ R d − φ −→ R d +1 φ −→ R (3)7s a complex of R -modules.We have φ · φ = (cid:2) z fa + x gb z fa z fa + x gb z fa + y d c x gb x gb + y d c z fa + z d d (cid:3) The fact that this is a null matrix is a routine exercise in syzygies of monomials as re-duced Koszul relations. Yet, the shape of φ will be of relevance later for rank and minorscomputation.The other composite φ · φ = a s + a s a t + a t + a t a u + a u + a u a v + a v b s + b s + b s b t + b t b u + b u ( d − × × ( d − c t + c t × ( d − × × ( d − × d u d v is a bit more delicate, but all calculations are straightforward. Claim 2.
The complex (3) is acyclic.For the argument one uses the Buchsbaum–Eisenbud acyclicity criterion. Obviously,rank φ = 1 and ht I ( φ ) ≥ . We next focus on φ , aiming at showing that rank φ ≥ d (hence, rank φ = 2 d ) andht I d ( φ ) ≥ . For this, we single out the following 2 d × d submatrices. A := a a d × ( d − d × b ( d − × b b ρ × ( d − × × ( d − × , formed with rows 1 , , . . . , d − , d + 1 and columns d, d + 1 , . . . , d −
1, where ρ = ( − z d − . . . x d − ) t is the first column of the rightmost degree block of φ , and B = a a a b ( d − × ( d − × d × ( d − c × d , formed with rows 1 , , . . . , d and columns d, d + 1 , . . . , d, d, . . . , d − det A = x d − det (cid:20) a a d × ( d − d × b ( d − × b b (cid:21) = x d − det (cid:2) a a (cid:3) det (cid:2) b b (cid:3) = x d − det (cid:20) x e d − ( d − × × ( d − y (cid:21) det − y · · · x − y · · · ... . . . . . . ... ... · · · x − y · · · x − y = x d − ( yx d − )(( − d − y d ) = ( − d − x d − y d +1 ∈ I d ( φ ) (4) det B = c det a det b = ( − z )( − d z ( d ( d − ( − d − ( z d − ) = z d ∈ I d ( φ ) (5) Therefore, one is through.Next get to φ , for which we want to prove that rank φ ≥ d − I d − ( φ ) ≥ φ = 2 d , then rank φ ≤ d −
2, so one derives the sought equality.The determinants of the following three (2 d − × (2 d −
2) submatrices of φ will beshown to form a regular sequence: S = s ( d − × u ( d − × × ( d − t × ( d − × ( d − × ( d − ( d − × ˜ u ˜ v ,T = (cid:20) s t u ( d − × ( d − × ( d − ( d − × ¯ u ¯ v (cid:21) and U = s ( d − × u ( d − × × ( d − t × ( d − v s ( d − × ( d − × ( d − ( d − × × ( d − t × ( d − × . Here • ˜ u and ˜ v are the submatrices obtained from u and v , respectively, by omitting thefirst row. • u and v are the submatrices obtained from u and v , respectively, by omitting thelast row.The calculation is straightforward: det S = t det s det (cid:2) ˜ u ˜ v (cid:3) = t det s det x − y · · · ... . . . . . . ... ... · · · x − y · · · x − y . . . x = x · x d − · x d − = x d − , (6) et T = det (cid:2) s t (cid:3) det (cid:2) ¯ u ¯ v (cid:3) = det (cid:20) − y e d − × ( d − ( d − × − y (cid:21) det − y · · · x − y · · · ... . . . . . . ... ... · · · x − y · · · x − y = y d − , (7)det U = ± det s · det ( d − × u ( d − × t × ( d − v t × ( d − × = ± det s det u det v det t = ± z d − z ( d − d − z d − z = ± z d ( d − . (8) This completes the argument on the acyclicity criterion, hence also the proof of item(a).(b) This is a consequence of (a) via a well-known argument (see, e.g., [24, Lemma 1.3]).(c) Let f , . . . , f d +1 denote the given set of generators of I. Then: R d − = I d − (by item (b))= R d − I d = R d − f + · · · + R d − f d +1 (9)Hence, by Proposition 2.1, P f / ∈ V ( I α ( M d, d +1 ,d − )) . In particular, V ( I α ( M d, d +1 ,d − )) is aproper subset of P N × · · · × P N . ✷ The previous example admits a “degree reparametrization” analogue:
Proposition 2.3.
Let d ′ ≥ be a integer and let I ′ ⊂ R = k [ x, y, z ] be the extension of theideal I = ( z ( x, y ) d − , x ( x, y ) d − , y d , z d ) by the endomorphism of k -algebras ζ : k [ x, y, z ] → k [ x, y, z ] x x d ′ , y y d ′ , z z d ′ . Then, (a)
The minimal graded free resolution of
R/I ′ has the form → R ( − ( dd ′ + 2 d ′ )) d − ⊕ R ( − ( dd ′ + 3 d ′ )) ⊕ R ( − (2 dd ′ )) d − → R ( − ( dd ′ + d ′ )) d − ⊕ R ( − ( dd ′ + 2 d ′ )) ⊕ R ( − (2 d ′ d − d ′ )) d → R ( − dd ′ ) d +1 → R → R/I ′ → (b) The decomposition structure of the socle of
R/I ′ is Soc(
R/I ′ ) = (cid:26) k ( − (4 d ′ − ⊕ k ( − (5 d ′ − , if d = 2 k ( − ( dd ′ + 2 d ′ − d − ⊕ k ( − ( dd ′ + 3 d ′ − ⊕ k ( − (2 dd ′ − d − , if d ≥ If, moreover, d ≥ then V ( I α ( M dd ′ , d +1 ,dd ′ + d ′ − )) is a proper subset of P N × · · · × P N . Proof. (a) Consider the matrix φ , φ and φ the matrix in the proof of the Proposition 2.2.Define ζ ( φ i ) ( i = 1 , ,
3) being the matrix obtained from φ by evaluating ζ in each of itsentries. Obviously, ζ ( φ ) · ζ ( φ ) = 0 and ζ ( φ ) · ζ ( φ ) = 0Hence, the sequence 0 → R d − ζ ( φ ) −→ R d − ζ ( φ ) −→ R d +1 ζ ( φ ) −→ R (10)is a complex.As shown in Proposition 2.2, there are 2 d × d submatrices A and B of φ such thatdet A = − z d ∈ I d ( φ ) and det B = x d − y d +1 ∈ I d ( φ )Thus, det ζ ( A ) = − z d d ′ ∈ I d ( ζ ( φ )) and det ζ ( B ) = x ( d − d ′ y (2 d +1) d ′ I d ( ζ ( φ ))Hence, rank ζ ( φ ) = 2 d and ht I d ( ζ ( φ )) ≥ . (11)The proof of Proposition 2.2 also guarantees the existence of (2 d − × (2 d −
2) submatrices
S, T, U of φ such that det S = y d , det T = ± z d ( d − , det U = x d − Thus, det ζ ( S ) = y dd ′ , det ζ ( T ) = ± z d ( d − d ′ , det ζ ( U ) = x (2 d − d ′ Hence, rank ζ ( φ ) = 2 d − I d ( ζ ( φ )) ≥ . (12)Therefore, the complex 10 is acyclic.By construction, the cokernel of ζ ( φ ) is R/I ′ . So,0 → R d − ζ ( φ ) −→ R d − ζ ( φ ) −→ R d +1 ζ ( φ ) −→ R → R/I ′ → R/I ′ . Finally, observing the degrees of the entries of the matrices ζ ( φ ) , ζ ( φ ) and ζ ( φ ) , the resolution (13) produces a minimal graded free resolution for R/I ′ as stated in the proposition. ✷ In order to overcome the hypothesis that d ≥ Proposition 2.4.
Given an integer d ′ ≥ , consider the following ideal of R = k [ x, y, z ] I = ( x d ′ , y d ′ , z d ′ , x d ′ y d ′ , xz d ′ − ) Then:
The minimal graded free resolution of
R/I has the form: → R ( − (4 d ′ + 1)) ⊕ R ( − (5 d ′ − → R ( − (2 d ′ + 1)) ⊕ R ( − d ′ ) ⊕ R ( − (4 d ′ − ⊕ R ( − d ′ ) → R ( − d ′ ) → R/I → (b) The structure decomposition of the socle of
R/I is Soc(
R/I ) = k ( − (4 d ′ − ⊕ k ( − (5 d ′ − (15)(c) V ( I α ( M d ′ , , d ′ − )) is a proper subset of P N × · · · × P N . Proof.
Once more, write φ = [ x d ′ y d ′ z d ′ x d ′ y d ′ xz d ′ − ] and introduce first and secondsyzygies candidates: φ = − y d ′ − z d ′ − x d ′ − z d ′ − xz d ′ − x y d ′ − y d ′ x d ′ − z d ′ − − z x d ′ − z d ′ x d ′ − y d ′ and φ = y d ′ z d ′ − z d ′ − x d ′ − y d ′ − y d ′ − x z x d ′ − . A straightforward calculation gives φ · φ = 0 and φ · φ = 0 . Thus, the sequence 0 → R φ → R φ → R φ → R is a complex.Now consider the following 4 × φ A = − z d ′ −
00 0 0 − z d ′ − z d ′ − − z x d ′ − z d ′ x d ′ − and B = x d ′ x − y d ′ x d ′ − z x d ′ − . As easily seen, det A = − z d ′ − and det B = − x d ′ . Therefore, rank φ = 4 and ht I ( φ ) ≥
2. 12n the other hand, for the following 3 × φ S = y d ′ x d ′ − y d ′ − y d ′ , T = z d ′ − z d ′ − z x d ′ − and U = x d ′ − y d ′ − x z x d ′ − one has det S = − y d ′ , det T = − z d ′ − and det U = x d ′ , forming a regular sequence.Thus, (a) holds.(b) and (c) follow easily as before. ✷ d ≥ are not Gorenstein One way of thinking about r linear forms f = { f , . . . , f r } ⊂ R = k [ x ] = k [ x , . . . , x n ] ofdegree d ≥ x d ] · Θ, where x d is the list of monomials of degree d and Θ denotes a (cid:0) d + n − d (cid:1) × r -matrix with entries in k .In the vein of looking at general such forms, Θ will be general. In the case where n = 3and r = 5, Θ is a 6 × Lemma 2.5.
Let Θ denote the generic × matrix over k . Let B denote the upper × submatrix of Θ and C = cof( B ) its matrix of cofactors. Set ∆ = det B and let ∆ , . . . , ∆ denote the entries of the product matrix L · C , where L is the last row of Θ . Then themultigraded polynomial D = det ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ − ∆ is non-vanishing. Proof.
Note that D is a polynomial in the entries of Θ. It suffices to prove that it evaluatesto a nonzero scalar at some scalar point. Choose P = (1 , , , , ,
1; 0 , , , , ,
1; 0 , , , , ,
0; 0 , , , , ,
0; 0 , , , , , , so Θ( P ) = Obviously, B ( P ) = cof( B )( P ) is the identity matrix.Therefore, ∆( B ( P )) = 1 and D ( P ) = det − = 1 . ✷ Our main goal is the following result. 13 heorem 2.6.
Let I ⊂ R = k [ x, y, z ] be an ideal of codimension generated by r generalforms of degree d . Then I is a Gorenstein ideal if and only if either r = 3 ( completeintersection ) or r = 5 and d = 2 ( five quadrics ) . Proof. ( ⇐ ) The space of parameters of r = 5 forms f ⊂ R = k [ x, y, z ] of degree 2 is P := P × P × P × P × P . The argument will consist in determining a non-empty openset U ⊂ P such that, in the terminology of Subsection 2.1, for any f with parameters P f ∈ U the ideal I = ( f ) is a codimension 3 Gorenstein ideal.Set U = P \ V ( I α ( M , , )) and U = P \ V (∆ · D ) where ∆ and D are as in Lemma 2.5.By the latter lemma, ∆ · D is a nonzero multigraded polynomial, hence U = ∅ .To see that U = ∅ as well, take any codimension 3 ideal I = ( f ) ⊂ R minimallygenerated by 5 quadrics. Then the length of ( x, y, z ) /I is 1, hence necessarily ( x, y, z ) ⊂ I : ( x, y, z ), showing that ( x, y, z ) ⊂ I . This means that R I = R . Consequently, P f ∈ U by Proposition 2.1, thus proving the contention. Claim.
For every f ∈ R × R × R × R × R with P f ∈ U := U ∩ U , the ideal I = ( f )is a codimension 3 Gorenstein ideal.First note that, since P f ∈ U then ( R/I ) j = 0 for every j ≥ . In particular, I is ( x, y, z )-primary and, as seen above, ( x, y, z ) /I has length 1. It follows that the socle I : ( x, y, z ) /I ⊂ R/I has only one generator in degree 2. To conclude that
R/I is Gorensteinit then suffices to prove that I : ( x, y, z ) contains no linear form.By our departing raison d’ˆetre, with the notation of Lemma 2.5, one has[ f f f f f ] = (cid:2) x xy xz y yz z (cid:3) Θ( P f ) . (16)Since P f ∈ U then ∆( P f ) = 0 . Multiplying both sides of (16) by C ( P f )∆( P f ) yields [ f f f f f ] C ( P f )∆( P f ) = (cid:2) x xy xz y yz z (cid:3) Θ( P f ) C ( P f )∆( P f )= (cid:2) x xy xz y yz z (cid:3) ∆ ( P f )∆( P f ) ∆ ( P f )∆( P f ) ∆ ( P f )∆( P f ) ∆ ( P f )∆( P f ) ∆ ( P f )∆( P f ) Thus, one gets the following generators I = ( x + ∆ ( P f )∆( P f ) z , xy + ∆ ( P f )∆( P f ) z , xz + ∆ ( P f )∆( P f ) z , y + ∆ ( P f )∆( P f ) z , yz + ∆ ( P f )∆( P f ) z ) (17) Now, assume ℓ = ax + by + cz is a linear form in Soc( R/I ) . Since xℓ = ax + bxy + cxz ∈ I ,using the above set of generators, yields (cid:18) ∆ ( P f )∆( P f ) a + ∆ ( P f )∆( P f ) b + ∆ ( P f )∆( P f ) c (cid:19) z ∈ I. But z does not belong to I , as otherwise I = ( x, y, z ) .14herefore, ∆ ( P f ) a + ∆ ( P f ) b + ∆ ( P f ) c = 0. By a similar token,∆ ( P f ) a + ∆ ( P f ) b + ∆ ( P f ) c = 0∆ ( P f ) a + ∆ ( P f ) b + ∆( P f ) c = 0Thus, ( a b c ) t is a solution of the linear system defined by the matrix ∆ ( P f ) ∆ ( P f ) ∆ ( P f )∆ ( P f ) ∆ ( P f ) ∆ ( P f )∆ ( P f ) ∆ ( P f ) − ∆( P f ) . However, since P f ∈ U then D ( P f ) = det ∆ ( P f ) ∆ ( P f ) ∆ ( P f )∆ ( P f ) ∆ ( P f ) ∆ ( P f )∆ ( P f ) ∆ ( P f ) − ∆( P f ) = 0 . This forces a = b = c = 0, as was to be shown.( ⇒ ) For the converse, one may right off assume that r ≥ f , . . . , f r be forms of degree d generating I . Since the socle degree of I is 2 d + d ′ − R d + d ′ − f + · · · + R d + d ′ − f r ( R d + d ′ − . Therefore, Proposition 2.1implies that P f ∈ V ( I α ( M d,r,d + d ′ − )) = V (( I α ( M (( r − / d ′ ,r, (( r − / d ′ + d ′ − ))) . However, if d ≥ V ( I α ( M d,r,d + d ′ − )) is aproper Zariski-closed subset of P N × · · · × P N . This shows that all Gorenstein ideals ofcodimension 3, of given degree d ≥ r ≥ I be generatedby general forms. ✷ Recall that a codimension 3 Gorenstein ideal I ⊂ R = k [ x , . . . , x n ] is generated by thePfaffians of an r × r skew-symmetric matrix S , with r = µ ( I ) odd. When I is equigenerated,say, in degree d ≥
1, then the columns of S must be homogeneous of some standard degree d i , ≤ i ≤ r . By the nature of each generator of I as a maximal Pfaffian of S , it immediatelyfollows that2 d = d + d + · · · + d r − + d r − = d + d + · · · + d r − + d r = · · · = d + d + · · · + d r − + d r . By an elementary argument, one derives the equalities throughout d = d = · · · = d r .It follows that 2 d = ( r − d ′ , where d ′ is the common values of the d i ’s, i.e., d = r − d ′ . (18)15hus, an equigenerated Gorenstein ideal I of codimension 3 complies with this relation,where r is the number of generators, d their common degree and d ′ is the degree of an entryof the skew–symmetric matrix defining I .It is often (if not mostly) the case that such a Gorenstein ideal is given by generatorsother than the Pfaffians. Therefore, it is important to underpin how much of the structurecomes out of these fringe integers.For convenience, a pair ( d, r ) of non-negative integers d ≥ , r ≥ virtual datum for Gorenstein ideals if it satisfies relation (18) for some integer d ′ ≥
1. Moreprecisely:
Definition 3.1. An equigenerated Gorenstein virtual datum in dimension n ≥ d, r ) of integers such that:(i) d ≥ r ≥ r − / d .Note that, since n ≥ r ≤ (cid:0) n + d − n − (cid:1) as it should be. The adjective ‘equigenerated’will often be omitted in the above terminology. When ( d, r ) is a virtual datum we will forconvenience call the integer 2 d/ ( r −
1) the skew-degree of the datum.A virtual datum is moreover proper if there exists a codimension 3 Gorenstein ideal in k [ x , . . . , x n ] with this datum, in which case the skew-degree is in fact the degree of anyentry in the alternating matrix whose maximal Pfaffians generated the ideal.We emphasize that if d ′ = 1 then r = 2 d + 1 – this case is the simplest one and will bereferred to as the linear case (i.e., the entries of the defining skew-symmetric matrix S arelinear forms). In particular, if the degree d of the generators is a prime number then eithera defining skew-symmetric matrix is linear or else I is a complete intersection, in which casethe matrix coincides with the matrix of the Koszul syzygies of the generators.The socle degree is easily established in terms of the virtual datum: Lemma 3.2.
Let I ⊂ R = k [ x, y, z ] denote an equigenerated codimension Gorensteinideal with virtual datum ( d, (2 d + d ′ ) /d ′ ) , where d ′ ≥ is the skew degree of I as introducedearlier. Then the socle degree of R/I is d + d ′ − . Proof.
It is well-known that, pretty generally, the socle degree of a graded ArtinianGorenstein quotient
R/I of a standard graded polynomial ring R = k [ x , . . . , x n ] is givenby D − n , where D is the last shift in the minimal graded R -resolution of R/I (see, e.g.,[24, Lemma 1.3]). In the present case, the resolution has length 3 and the first syzygieshave shifted degree d + d ′ . Therefore, D = d + d + d ′ = 2 d + d ′ , hence the socle degree is asstated. ✷ Next is a list of examples of equigenerated Gorenstein ideals of low degrees in k [ x, y, z ]in characteristic zero. Besides stressing the presence of the virtual datum, they may serveas motivation for later results and questions. The emphasis is on the virtual datum ( d, r ),with the eclipsed skew-degree d ′ = 2 d/ ( r −
1) behind the scenes.
Example 3.3. ( d = 2 , µ ( I ) = 5)( x , y , z ) : x y + x z + y z = ( xy, xz, yz, x − z , y − z ) . k ) = 0, in the notation of [21], one has I (2) = ( x , y , z , xz − yz, xy − yz ),whereas the ideal with 5 × x , y , xz, yz, xy + z ).Now, by [15, Theorem 3.1], over k = R , there are only three orbits under GL (3 , R ). Arethese representatives of each orbit over R ?) Example 3.4. ( d = 3 , µ ( I ) = 7)( x , y , z ) : x + y + z = ( x , y , z , xyz, x ( y − z ) , y ( x − z ) , z ( x − y ) . There is no outset reason not to consider colon with respect to more diversified regularsequences. Take the following three quadrics: { q = x − yz, q = y − xz, q = z + xy } ⊂ k [ x, y, z ] . One can check that these form a regular sequence if char ( k ) = 2 (note the exceptionalsign). Then: Example 3.5. ( d = 4 , µ ( I ) = 5)( q , q , q ) : x + y + z . A different choice with same virtual datum is( x , y , z ) : ( x + y + z ) . (The latter is a special case of the following situation: d = 2 b an even number and r = d + 1 = 2 b + 1 – hence, d ′ = 2 – with corresponding Gorenstein ideal given by( x d +1 , y d +1 , z d +1 ) : ( x + y + z ) d +1 . Our later development suggests that this case hasgeneral well-understood features.) Example 3.6. ( d = 4 , µ ( I ) = 9)( q , q , q ) : x + y + z . Another choice giving the same virtual datum is( x , y , z ) : ( x + y + z ) a case to be considered below in a more general scope.Yet another choice is ( x , y , z ) : ( x y + x z + y z ) . This multiplicity of exampleseven for such small values of the virtual datum is a prelude of how difficult might be theclassification under the action of GL(3 , k ). Example 3.7. ( d = 6 , µ ( I ) = 5)Turning around the examples so far, we give instead the skew-symmetric matrix whosemaximal Pfaffians generate an ideal of codimension 3 with the stated datum: N = xyz y x x y − xyz z y z − y − z z x − x − y − z z − x y − z − x − z . xample 3.8. ( d = 6 , µ ( I ) = 7)( x , y , z ) : ( x + y + z ) . This example is a reparametrization of Example 3.4. As will be seen, reparametrizationhas some role in the subject.
For any degree d ≥
1, a corresponding m -primary Gorenstein ideal with linear skew–symmetric matrix (hence, µ ( I ) = 2 d + 1) is exemplified by the ideal I ( d ) := ( x d , y d , z d ) :( x + y + z ) d − (cf. [21, Proposition 7.24]). A drawback of this example is that it may requirechar( k ) = 0 – or sufficiently high characteristic – in order to get an equigenerated ideal. Forexample, if char( k ) = 2 then x + y + z ∈ I (2). By and large, a typical behavior will oftenchange in positive characteristic.In this part one proves, in a characteristic-free way, that any virtual datum in dimension n ≥ k [ z , . . . , z N ] be a standard graded polynomial ring over an infinite field k . Givenan integer p ≥
2, consider the following injective k -algebra map ζ p : k [ z , . . . , z N ] → k [ z , . . . , z N ] , z i z pi (1 ≤ i ≤ N ) Lemma 3.9.
Let > be a monomial order on k [ z , . . . , z N ] . Then, in > ( ζ p ( f )) = ζ p (in > ( f )) for every form f . Proof.
We can write f = a z v + · · · + a s z v s where each a i = 0 for each i and z v > z v j for each j ≥ . In particular, in > ( f ) = z v . Wehave ζ p ( f ) = a ( z v ) p + · · · + a s ( z v s ) p By the properties of a monomial order one has ( z v ) p > ( z v j ) p for each j ≥ . Thus, inparticular, in > ( ζ p ( f )) = ( z v ) p = ζ p ( z v ) = ζ p (in > ( f )) . ✷ Let r be a odd integer. Denote by X the r × r generic skew-symmetric matrix x , · · · x ,r − x , · · · x ,r ... ... . . . ... − x ,r − x ,r · · · Denote by B the polynomial ring over k in the variables of X. emma 3.10. Let I be the ideal of B generated by the ( r − -Pfaffians of X and ζ p ( I ) theextension of I by the epimorphism ζ p . Then, ζ p ( I ) is a codimension Gorenstein ideal of B with structural skew–symmetric matrix ζ p ( X ) = x p , · · · x p ,r − x p , · · · x p ,r ... ... . . . ... − x p ,r − x p ,r · · · Proof.
Since homomorphism of k -algebras preserves determinants, then the ideal ζ p ( I ) isgenerated by the ( r − ζ p ( X ) . Thus, in order to conclude the proof it sufficesto show that ht ζ p ( I ) ≥ . By [13, Theorem 5.1] there is a monomial order > on B such that the ( r − X constitute a Gr¨obner base of I. In particular, if f , . . . , f r are the ( r − I then in > ( I ) = (in > ( f ) , . . . , in > ( f r )) . Thus,in > ( ζ p ( I )) ⊃ (in > ( ζ p ( f )) , . . . , in > ( ζ p ( f r )))= ( ζ p (in > ( f )) , . . . , ζ p (in > ( f r ))) (by Lemma 3.9) (19)= (in > ( f ) p , . . . , in > ( f r ) p )Thus, ht ζ p ( I ) = ht in > ( ζ p ( I )) ≥ ht (in > ( f ) p , . . . , in > ( f r ) p ) (by (19))= ht (in > ( f ) , . . . , in > ( f r )) (20)= ht in > ( I )= ht I = 3 . ✷ Proposition 3.11.
Let n ≥ be a integer. Then every virtual datum ( d, r ) in dimension n is proper. Proof.
Denote by d ′ the skew degree of the virtual datum ( d, r ) . Define u = (cid:0) r (cid:1) − n ≤ (cid:0) r (cid:1) − . The proof is by induction on u. If u = 0, by Lemma 3.10 the ideal generated by the ( r − x d ′ , · · · x d ′ ,r − x d ′ , · · · x d ′ ,r ... ... . . . ... − x d ′ ,r − x d ′ ,r · · · is a Gorenstein ideal with datum ( d, r ) . Now, suppose that the result is true for a certain1 ≤ u < (cid:0) r (cid:1) − . Then there is an r × r skew-symetric matrix Ψ = ( g i,j ) whose entries g i,j areforms of degree d ′ in k [ x , . . . , x n ] and I = Pf r − (Ψ) is a codimension 3 Gorenstein ideal with19atum ( d, r ) . Since one can assume that r is fixed, one can argue by descending inductionon n instead. Thus, we are assuming that n >
3. Since I is an unmixed homogeneous idealof codimension 3, then by a homogeneous version of the prime avoidance lemma, there is alinear form ℓ ∈ k [ x , . . . , x n ] that is R/I -regular. Without loss of generality one can suppose ℓ = x n − P n − i =1 α i x i . Now consider the following surjective k -algebra homomorphism: π : k [ x , . . . , x n ] ։ k [ x , . . . , x n − ] , x i x i (1 ≤ i ≤ n − , x n n − X i =1 α i x i . Since determinants commute with ring homomorphisms, π induces a k -algebra isomorphism k [ x , . . . , x n ] / (Pf r − (Ψ) , ℓ ) ≃ k [ x , . . . , x n − ] / Pf r − ( e Ψ)where e Ψ = ( π ( g ij )) . Since ℓ is ( k [ x , . . . , x n ] /I )-regular then k [ x , . . . , x n − ] / Pf r − ( e Ψ) is aGorenstein ring of codimension( n − − dim k [ x , . . . , x n − ] / Pf r − ( e Ψ) = ( n − − ( n −
4) = 3 . Thus, Pf r − ( e Ψ) is a codimension 3 Gorenstein ideal in k [ x , . . . , x n − ] with datum ( d, r ) . ✷ In this brief part one looks at the behavior of some algebras associated to a codimension 3equigenerated Gorenstein ideal I ⊂ R = k [ x, y, z ]. We will particularly focus on the Reesalgebra R ( I ) ≃ R [ It ] ⊂ R [ t ], the associated graded ring gr I ( R ) = R ( I ) /I R ( I ) and thefiber cone algebra F ( I ) = R ( I ) / m R ( I ), where m := ( x, y, z ). The nature of the associatedgraded ring for Artinian Gorenstein rings in any dimension has been considered earlier byIarrobino ([17]).It is important to observe that the so-called condition ( G ) is automatic since the idealis m -primary. Therefore, some features in this section are most certainly contained in themonumental [23]. Still, it may be useful to have direct elementary proofs of the resultsbelow, where G is not directly used.The first result is quite more general. Proposition 3.12.
Let R = k [ x , . . . , x n ] be a standard graded polynomial ring over afield and let m be its maximal homogeneous ideal. Let I be a d -equigenerated homogeneous m -primary ideal. Given an integer m ≥ such that I m = m dm , then the following hold: (a) I m = m md for every m ≥ m . (b) The reduction number of I is at most max { m , r ( m d ) } , where r ( m d ) denotes the re-duction number of m d . (c) The ( regular ) rational map F : P n − P µ ( I ) − defined by a set of forms spanning I d is birational onto the image. (d) The Rees algebra R ( I ) satisfies the condition R of Serre. (e) depth gr I ( R ) = 0 . roof. (a) One has I m ⊂ m d I m − ⊂ m d m ( m − d = I m , hence, I m = m d I m − . Inducting on m ≥ m , I m +1 = I m +1 − m I m = I m +1 − m m d I m − = I m m d = m ( m +1) d . (b) Let J ⊂ I be a homogeneous minimal reduction. Since m d is the integral closure of I , then J is also a minimal reduction of m d . Setting N = max { m , r ( m d ) } , one has: I N +1 = ( m d ) N +1 by (a)= J ( m d ) N because J is a minimal reduction of m d = J I N by (a)(c) By (a), the Hilbert polynomial HP ( F ( I ) , m ) of the special fiber F ( I ) is HP ( F ( I ) , m ) = (cid:18) md + n − n − (cid:19) = d n − ( n − m n − + lower degree terms of m. Hence, the multiplicity e ( F ( I )) of F ( I ) is d n − . On the other hand, by [27, Theorem 6.6(a)] the degree deg( F ) of the rational map F isdeg( F ) = d n − e ( F ( I ))Thus, deg( F ) = 1, as asserted.(d) Consider the Hilbert-Samuel polynomial ( m >> λ ( R/I m +1 ) = e ( I ) (cid:18) n + mn (cid:19) − e ( I ) (cid:18) n + m − n − (cid:19) + lower degree terms of m and the Hilbert polynomial λ ( R/I m +1 ) = e ( I ) (cid:18) n + mn (cid:19) − e ( I ) (cid:18) n + m − n − (cid:19) + lower degree terms of m where I m +1 denotes the integral closure of I m +1 . By (a), I m = I m for every m ≥ m . Thus,in particular, e ( I ) = e ( I ) . Hence, by [14, Proposition 3.2], R ( I ) satisfies the condition R of Serre.(e) By (a), one has an exact sequence0 −→ R ( I ) −→ R ( m d ) −→ C −→ , with C a module of finite length. In particular, depth C = 0 . Since R ( m d ) is Cohen–Macaulay, then depth R ( I ) = depth C + 1 = 1.Now, clearly depth gr I ( R ) ≤ depth R ( I ) = 1. Supposing that depth gr I ( R ) >
0, let a ∈ I \ I be such that its image in I/I ⊂ gr I ( R ) is a regular element. Then one has anexact sequence 0 → gr I ( R )( − → R ( I ) /a R ( I ) → R R/ ( a ) ( I/ ( a )) → a is regular on R ( I ) then the middle term has depthzero, while the rightmost term – being a Rees algebra over a Cohen–Macaulay ring ofdimension ≥ I ( R ) ≃ gr I ( R )( −
1) has depth zero;a contradiction. ✷ For the main result in this part recall the notion of a syzygetic ideal I ⊂ R as beingone such that the natural surjection S R ( I ) ։ R R ( I ) is an isomorphism in degree ≤
2. Inparticular, for such an ideal, I coincides with the second symmetric power of I , hence thenumber of generators µ ( I ) is given by (cid:0) µ ( I )+12 (cid:1) .In the ternary case one can bring up the special fiber. Theorem 3.13.
Let I be a d -equigenerated m -primary homogeneous ideal in the standardpolynomial ring R = k [ x, y, z ] , with d ≥ . If I is syzygetic and minimally generated by d + 1 forms, the following hold: (a) I = m d . (b) I m = m md for every m ≥ . (c) The reduction number of I is . (d) The ( regular ) rational map F : P P d defined by a set of forms spanning I d isbirational onto the image. (e) The Rees algebra R ( I ) satisfies the condition R of Serre. (f) depth gr I ( R ) = 0 . (g) The special fiber F ( I ) is not Cohen-Macaulay. Proof. (a) Since I is syzygetic then µ ( I ) = (cid:18) µ ( I ) + 12 (cid:19) = (cid:18) d + 22 (cid:19) = µ ( m d ) . Thus, I ⊂ m d is a inclusion of homogeneous ideal generated in fixed degree 2 d having thesame minimal number of homogenous generators. Hence, I = m d . It remains now to prove items (c) and (g) because the others follow exactly as in Propo-sition 3.12.(c) By (a) and Proposition 3.12, one has r ( I ) ≤ . On the other hand, since I is syzygeticone has 2 ≤ r ( I ) . Hence, r ( I ) = 2 . (g) Suppose to the contrary. Then, by [10, Proposition 1.2], the reduction number r ( I )is the Castelnuovo-Mumford regularity reg( F ( I )) of F ( I ) . By (c), the latter is 2. But since I is syzygetic, the defining ideal of F ( I ) over S := k [ T , . . . , T d +1 ] admits no forms ofdegree 2, hence is generated in the single degree 3. In particular, the minimal graded freeresolution of F ( I ) over S is linear:0 → S ( − N + 1) β N − → · · · → S ( − β → S.
22y [16, Theorem 1.2], the multiplicity of the special fiber F ( I ) is e ( F ( I )) = (cid:18) µ ( I ) − (cid:19) = (cid:18) d (cid:19) . Now consider the rational map F : P P d defined by the given generators of I in degree d , and let deg( F ) denote the degree of F . Since I is equigenerated then F ( I ) is isomorphicto the k -subalgebra of R generated by the vector space I d , while the latter is up to degreenormalization the homogeneous defining ideal of the image of F . Then, by [27, Theorem6.6 (a)] one has (cid:18) d (cid:19) deg( F ) = e ( F ( I )) deg( F ) = d . Since F is birational, 2 d − d , which is absurd for d ≥ ✷ Corollary 3.14. (char( k ) = 2) Let I denote a codimension homogeneous Gorenstein idealin k [ x, y, z ] with datum ( d, d + 1) , where d ≥ . Then all assertions of Theorem 3.13 holdtrue. Proof.
Since char( k ) = 2, then I is syzygetic ([12, Proposition 2.8]). ✷ For the non-linear case, one has the following:
Proposition 3.15.
Let I ⊂ R = k [ x, y, z ] be a codimension Gorenstein ideal with datum ( d, r ) and skew degree d ′ . Let F : P P r − be the rational map defined by the linearsystem I d . If the reduction number of I is at most and F ( I ) is Cohen-Macaulay then: (a) ( r − divide d ′ . (b) If r > then F is not birational onto the image. Proof. (a) Since I is syzygetic, the assumption implies that the reduction number of I is exactly 2. Since F ( I ) is Cohen-Macaulay then the same argument as in the proof ofTheorem 3.14 (c) yields e ( F ( I )) = (cid:0) r − (cid:1) . Again, by [27, Theorem 6.6], (cid:0) r − (cid:1) deg( F ) = d . By definition, d = ( r − d ′ /
2, hence2( r −
2) deg( F ) = ( r − d ′ . (21)Since gcd { ( r − , ( r − } = 1 then ( r −
2) divides d ′ , as desired.(b) Since ( r − / > F ) > . Hence, F is not birational. ✷ ( x m , . . . , x mn ) -colon problem It is known (see [1, Proposition 1.3]) that any homogeneous Gorenstein ideal of codimension n in k [ x ] = k [ x , . . . , x n ] can be obtained as a colon ideal ( x m , . . . , x mn ) : f , for some integer m ≥ f . In this section one deals with some of the main questions regardingthis representation.It is first established under which condition the form f is uniquely determined and whatis its degree in terms of m and the socle degree of I . Then one proves that f can be23etrieved from I by taking the so-called (socle-like) Newton dual of a minimal generator ofthe Macaulay inverse of I .Then one gives conditions under which the Gorenstein ideal I is equigenerated in termsof the exponent m and the form f . We solve this problem in the case where I has linearresolution0 → R ( − d − n + 2) → R ( − d − n + 2) b n − → · · · → R ( − d − b → R ( − d ) b → R where b = µ ( I ). The general case is still open, justifying perhaps calling it more particularlythe colon problem .For convenience, call f a directrix form (of I ) associated to the regular sequence { x m , . . . , x mn } . The Macaulay–Matlis duality meets yet another version in terms of the Newton polyhedronnature of the homogeneous forms involved so far.For this, recall the notion of the Newton (complementary) dual of a form f ∈ k [ x ] = k [ x , . . . , x n ] in a polynomial ring over a field k (see [5], [7]). Namely, let A denote the logmatrix of the constituent monomials of f (i.e, the nonzero terms of f ). This is the matrixwhose columns are the exponents vectors of the nonzero terms of f in, say, the lexicographicordering. It is denoted N ( f ).The Newton dual log matrix (or simply the Newton dual matrix) of the Newton logmatrix N ( f ) = ( a i,j ) is the matrix \ N ( f ) = ( α i − a i,j ) , where α i = max j { a i,j } , with1 ≤ i ≤ n and j indexes the set of all nonzero terms of f .In other words, denoting α := ( α · · · α n ) t , one has \ N ( f ) = [ α | · · · | α ] ( n +1) × r − N ( f ) , where r denotes the number of nonzero terms of f . The vector α is called the directrixvector of N ( f ) (or of f by abuse).We note that taking the Newton dual is a true duality upon forms not admitting mono-mial factor, in the sense that, for such a form f , \\ N ( f ) = N ( f ) holds.We define the Newton dual of f to be the form ˆ f whose terms are the ordered monomialsobtained form \ N ( f ) affected by the same coefficients as in f .Our next result asserts that directrix forms and Macaulay inverse generators obey aduality in terms of the above Newton dual. Given a directrix form f associated to theregular sequence { x m , . . . , x mn } – i.e., ( x m , . . . , x mn ) : f = I – it will typically admit monomialfactors. In order to fix this inconvenient, one redefines the socle-like Newton dual of suchdirectrix form by taking as directrix vector ν := ( m − · · · m − t . Proposition 4.1.
Let I ⊂ R = k [ x ] = k [ x , . . . , x n ] be a homogeneous codimension n Gorenstein ideal with socle degree s . Given an integer m ≥ , suppose that I admits adirectrix form f associated to the regular sequence { x m , . . . , x mn } . Then: (i) f is a degree n ( m − − s form uniquely determined, up to a scalar coefficient, by thecondition that no nonzero term of f belongs to the ideal ( x m , . . . , x mn ) . The socle-like Newton dual of f is a minimal generator of the Macaulay inverse to I ( having dual degree s ) , and its socle-like Newton dual retrieves f . Proof.
Suppose f = X | α | =deg f a α x α . Then, the socle-like Newton dual of f is ˆ f = X α a α y ˆ α , where ˆ α := ν − α (in particular, deg ˆ f = n ( m − − deg f ). Given a homogeneous polynomial h = X | β | =deg h b β x β ∈ R one has: h f = X | γ | =deg f +deg h X α + β = γ a α b β x γ and h ˆ f = X | γ | =deg f +deg h X ˆ α − β =ˆ γ a α b β y ˆ γ with ˆ γ = ν − γ. In particular, for every γ, the coefficient of x γ as a term in h f is equal tothe coefficient of y ˆ γ as a term of h ˆ f . Moreover, the i th coordinate of γ is larger than s ifand only if the i th coordinate of ˆ γ is negative. Thus, h ∈ ( x m , . . . , x mn ) : f ⇔ for every X α + β = γ a α b β = 0 , γ has a coordinate larger than m ⇔ for every X ˆ α − β =ˆ γ a α b β = 0 , ˆ γ has a negative coordinate (22) ⇔ h ˆ f = 0 ⇔ h ∈ Ann(ˆ f )Therefore, I = Ann(ˆ f ) , that is, ˆ f is a minimal generator of the Macaulay inverse to I . Byconstruction, one has deg ˆ f = n ( m − − deg f . On the other hand, it is well known thatthe degree of a minimal generators of the Macaulay inverse to I is the socle degree of I ,i.e., deg ˆ f = s . Therefore, deg f = n ( m − − s. Since ˆ f is uniquely determined, up to ascalar coefficient, the form f is uniquely determined as well, up to a scalar coefficient, bythe condition that no nonzero term of f belongs to the ideal ( x m , . . . , x mn ). Thus, assertion(i) follows.Assertion (ii) follows from the above. ✷ Remark 4.2.
Item (i) of Proposition 4.1 is stable under a change of coordinates. In otherwords, it holds true replacing the sequence { x m , . . . , x mn } by a sequence { ℓ m , . . . , ℓ mn } , where { ℓ , . . . , ℓ n } are independent linear forms. Thus, if I is a homogeneous codimension n Gorenstein ideal such that ( ℓ m , . . . , ℓ mn ) : f = I , for some form f ∈ R , then f is uniquelydetermined, up to a scalar coefficient, by the condition that no nonzero term of f , writtenas a polynomial in ℓ , . . . , ℓ n , belongs to the ideal ( ℓ m , . . . , ℓ mn ). In this section we characterize when I is a equigenerated codimension n Gorenstein idealwith linear resolution in terms of the exponent m and the form f ∈ R = k [ x ] = k [ x , . . . , x n ] . The preliminaries remain valid in arbitrary characteristic, but characteristic zero is calledupon in item (ii) of Proposition 4.5 below. 25et e, e ′ be positive integers and let f = P | α | = e a α x α ∈ R e and g = P | β | = e ′ b β x β ∈ R e ′ be forms. Given an integer m ≥
1, write g f = X x γ / ∈ ( x m ,...,x mn ) X α + β = γ a α b β x γ + X x γ ∈ ( x m ,...,x mn ) X α + β = γ a α b β x γ , (23)where γ ∈ N n is a running n -tuple. To this writing associate a matrix M e,e ′ ,m whose rowsare indexed by the n -tuples γ such that | γ | = e + e ′ and whose columns are indexed by the n -tuples β such that | β | = e ′ . The entries of the matrix are specified as follows:the ( γ, β )-entry of M e,e ′ ,m = (cid:26) , if some coordinate of γ − β is < a α , if each coordinate of α = γ − β is ≥ χ denote the row matrix [ x γ ] with the monomial entries x γ / ∈ ( x m , . . . , x mn ),and let b stand for the column matrix whose entries are the coefficients b β of g. Then equality(23) can be rewritten in the shape g f = χ · M e,e ′ ,m · b + X x γ ∈ ( x m ,...,x mn ) X α + β = γ a α b β x γ . (24)It is important to observe that the matrix M e,e ′ ,m depends only on the integers e, e ′ and m , and not on the details of g .From this, it follows immediately: Lemma 4.3. g ∈ I = ( x m , . . . , x mn ) : f if and only if M e,e ′ ,m · b = 0 . In particular, I e ′ = { } if and only if rank M e,e ′ ,m = (cid:0) e ′ + n − n − (cid:1) . To tie up the ends, consider the parameter map R e → P ( e + n − n − ) − , f = X | α | = e a α x α P f = ( a ( e,..., : · · · : a (0 ,...,e ) )in the notation of Subsection 2.1. Let { Y e,..., , . . . , Y ,...,e } denote the coordinates of P ( e + n − n − ) − and let MG e,e ′ ,m stand for the matrix whose entries are obtained by replacing each a α in M e,e ′ ,m by the corresponding Y α . Theorem 4.4.
Let m ≥ be a integer and let f ∈ R = k [ x , . . . , x n ] be a form. Thefollowing are equivalent: (i) I = ( x m , . . . , x mn ) : f is a codimension n equigenerated Gorenstein ideal with linearresolution. (ii) The integer s := n ( m − − deg f is even and rank M deg f ,s/ ,m = (cid:0) s/ n − n − (cid:1) . (iii) The integer s := n ( m − − deg f is even and P f is a point in the Zariski open set P ( e + n − n − ) − \ V ( I k ( MG e,s/ ,m )) , with k := (cid:0) s/ n − n − (cid:1) and e = deg f . roof. (i) ⇒ (ii) Suppose that I is equigenerated in degree d. Since I has linear resolutionthen the socle degre of I is 2 d − . Thus, by the Proposition 4.1, s = 2 d − . In particular, s isan even integer. On the other hand, since I is generated in degree d then I s/ = I d − = { } . Hence, by the Lemma 4.3, rank M deg f ,s/ ,m = (cid:0) s/ n − n − (cid:1) . (ii) ⇒ (i) We claim that I is codimension n Gorenstein ideal generated in degree t = s/ . The ideal I is Gorenstein of codimension n because it is the link of the homo-geneous almost complete intersection J = ( x m , . . . , x mn , f ) with respect to the completeintersection of pure powers ( x m , . . . , x mn ) . By Proposition 4.1, the socle degree of I is s. Thus, (
R/I ) t − = ( R/I ) s +1 = { } . On the other hand, since rank M deg f ,s/ ,m = (cid:0) s/ n − n − (cid:1) , one has, by Lemma 4.3, I t − = I s/ = { } . Since I is a codimension n Gorenstein ideal and(
R/I ) t − = { } and I t − = 0 it follows from [21, Proposition 1.8] that I is generated indegree t and has linear resolution.(ii) ⇔ (iii) This is a mere language transcription. ✷ The question remains as to when the Zariski open set P ( e + n − n − ) − \ V ( I k ( MG e,s/ ,m ))is nonempty, where k := (cid:0) s/ n − n − (cid:1) and e = deg f . The next result determines all pair ofintegers m, e ≥
1, with even s = n ( m − − e , for this to be the case when f = ( x + · · · + x n ) e . Proposition 4.5. (char( k ) = 0) Let m, e ≥ integers such that s = n ( m − − e is even.Set d := s/ . (i) If m < d then P ( e + n − n − ) − \ V ( I k ( MG e,s/ ,m )) = ∅ . (ii) If m ≥ d then I = ( x m , . . . , x mn ) : ( x + · · · + x n ) e is a codimension n Gorensteinideal generated by forms of degree d with linear resolution. In particular, P ( e + n − n − ) − \ V ( I k ( MG e,s/ ,m )) is a dense open set. Proof. (i) We claim that there is no form f of degree e such that I = ( x m , . . . , x m ) : f isa equigenerated codimension n Gorenstein ideal with linear resolution. In fact, otherwise I would be an ideal generated in degree d with ( x m , . . . , x mn ) ⊂ I – an absurd. Hence, byTheorem 4.4, P ( e + n − n − ) − \ V ( I k ( MG e,s/ ,m )) = ∅ . (ii) We mimic the argument of [21, Proposition 7.24]. Namely, by applying [21, Propo-sition 1.8], it is sufficient to show that R d − ⊂ I and I d − = { } . Clearly, R n ( m − ⊂ ( x m , . . . , x mn ) . Moreover, ( x + · · · + x n ) e R d − ⊂ R d − e = R n ( m − . Hence, R d − ⊂ I. On the other hand, the initial degree of I is d as a consequence of theLefschetz like result of R. Stanley, as proved in [26, Theorem 5]. Therefore, I d − = { } , aswas to be shown. In particular, for f = ( x + · · · + x n ) e Theorem 4.4 gives P f ∈ P ( e + n − n − ) − \ V ( I k ( MG e,s/ ,m )) . ✷ Remark 4.6.
The only place where one needs characteristic zero above is in the use of [26,Theorem 5] – for a different proof of this typical charming result of characteristic zero, see[3, Th´eor`eme, Appendix]. It is reasonable to expect that the above proposition be valid inarbitrary characteristic. 27 .3 The pure power gap
Let ℓ = { ℓ , . . . , ℓ n } ∈ R = k [ x , . . . , x n ] be a regular sequence of linear forms and let I ⊂ R be a homogeneous codimension n Gorenstein ideal with socle degree s. Denote by m ( I, ℓ ) theleast index m such that { ℓ m , . . . , ℓ mn } ⊂ I. Since R s +1 = I s +1 , then m ( I, ℓ ) ≤ s + 1 . The purepower gap of I with respect to the regular sequence ℓ is g ( I, ℓ ) := s +1 − m ( I, ℓ ) . The absolutepure power gap of I (or simply, the pure power gap of I ) is g ( I ) := s + 1 − min ℓ { m ( I, ℓ ) } . To start one has the following basic ring-theoretic result:
Lemma 4.7.
Let m , . . . , m n ≥ be integers and f a form in R = k [ x , . . . , x n ] . Then ( ℓ m , . . . , ℓ m n n ) : f = ( ℓ m , . . . , ℓ m i +1 i , . . . , ℓ m n n ) : ℓ i f for every ≤ i ≤ n. In particular, ( ℓ m , . . . , ℓ m n n ) : f = ( ℓ m + k , . . . , ℓ m i + ki , . . . , ℓ m n + kn ) : ( ℓ · · · ℓ n ) k f for each k ≥ . Proof.
One can assume that i = 1 . The inclusion ( ℓ m , . . . , ℓ m n n ) : f ⊂ ( ℓ m +11 , . . . , ℓ m n n ) : ℓ f is immediate. Thus, consider h ∈ ( ℓ m +11 , . . . , ℓ m n n ) : ℓ f . Then, ℓ f h = p ℓ m +11 + · · · + p n ℓ m n n for certain p , . . . , p n ∈ R. In particular, ℓ divide p ℓ m + · · · + p n ℓ m n n . We can write p i = ℓ q i + r i , for each 2 ≤ i ≤ n, where r , . . . , r n are polynomials in k [ ℓ , . . . , ℓ n ] . Thus, p ℓ m + · · · + p n ℓ m n n = q ℓ ℓ m + · · · + q n ℓ ℓ m n n + r ℓ m + · · · + r n ℓ m n n . Since ℓ divides p ℓ m + · · · + p n ℓ m n n and r ℓ m + · · · + r n ℓ m n n ∈ k [ ℓ , . . . , ℓ n ] then p ℓ m + · · · + p n ℓ m n n = q ℓ ℓ m + · · · + q n ℓ ℓ m n n . Thus, f h = p ℓ m + q ℓ m + · · · + q n ℓ m n n , that is, h ∈ ( ℓ m , . . . , ℓ m n n ) : f . Therefore, ( ℓ m , . . . , ℓ m n n ) : f = ( ℓ m +11 , . . . , ℓ m n n ) : ℓ f asstated. ✷ To see an application, recall from Remark 4.2 that if I is a homogeneous codimension n Gorenstein ideal such that ( ℓ m , . . . , ℓ mn ) : f = I , where ℓ , . . . , ℓ n are linear forms, then f is uniquely determined, up to a scalar coefficient, by the condition that no nonzero term of f , written as a polynomial in ℓ , . . . , ℓ n , belongs to the ideal ( ℓ m , . . . , ℓ mn ). Proposition 4.8.
Let I ⊂ R be a homogeneous codimension n Gorenstein ideal with socledegree s. Suppose that as above, ℓ , . . . , ℓ n are linear forms such that I = ( ℓ s +11 , . . . , ℓ s +1 n ) : f with f uniquely determined, up to a scalar coefficient, by the condition that no nonzeroterm of f belongs to the ideal ( ℓ s +11 , . . . , ℓ s +1 n ) . Then, g ( I, ℓ ) is the largest index such that ( ℓ · · · ℓ n ) g ( I, ℓ ) divides f . roof. Denote m := m ( I, ℓ ) and g := g ( I, ℓ ) . Then ( ℓ m , . . . , ℓ m n ) : I is an almostcomplete intersection J = ( ℓ m , . . . , ℓ m n , f ), for some form f ∈ R . Since R = k [ ℓ , . . . , ℓ n ],one can write f as a polynomial in these linear forms and get rid of the terms belongingto the ideal ( ℓ m , . . . , ℓ m n ). This way, the latter is part of a minimal set of generators of J .Therefore, ( ℓ m , . . . , ℓ m n ) : J = ( ℓ m , . . . , ℓ m n ) : f is Gorenstein and I = ( ℓ m , . . . , ℓ m n ) : f .By Lemma 4.7 one has I = ( ℓ m , . . . , ℓ m n ) : f = ( ℓ m + g , . . . , ℓ m + g n ) : ( ℓ · · · ℓ n ) g f = ( ℓ s +11 , . . . , ℓ s +1 n ) : ( ℓ · · · ℓ n ) g f (25)Consider f = X | α | =deg f a α ℓ α · · · ℓ α n n . Then, ( ℓ · · · ℓ n ) g f = X | α | =deg f a α ℓ α + g · · · ℓ α n + g n . For each nonzero a α , one has α i ≤ m − ≤ i ≤ n. Hence, α i + g ≤ m + g − s for each 1 ≤ i ≤ n. Thus, no nonzero term of ( ℓ · · · ℓ n ) g f belongs tothe ideal ( ℓ s +11 , . . . , ℓ s +1 n ) . Then, since f is uniquely determined, up to a scalar coefficient,by I = ( ℓ s +11 , . . . , ℓ s +1 n ) : f and the condition that no nonzero term of f belongs to theideal ( ℓ s +11 , . . . , ℓ s +1 n ), one has f = λ ( ℓ · · · ℓ n ) g f for some nonzero λ ∈ k . Hence, ( ℓ · · · ℓ n ) g divides f . Finally, one asserts that g is the largest index with this property, a claim that is obviousif m = 1 , because in this case f is a nonzero scalar. Thus, suppose m ≥ . If g is not thelargest index such that ( ℓ · · · ℓ n ) g divides f then ℓ · · · ℓ n divides f . Hence, by Lemma 4.7, I = ( ℓ m , . . . , ℓ m n ) : f = ( ℓ m − , . . . , ℓ m − n ) : f ℓ · · · ℓ n , so, { ℓ m − , . . . , ℓ m − n } ⊂ I, contradicting that m is least such that { ℓ m , . . . , ℓ m n } ⊂ I. ✷ References [1] D. Buchsbaum and D. Eisenbud, Remarks on ideals and resolutions, Istituto Nazionaledi Alta Matematica, Symposia Mathematica, Volume XI, Bologna (1973). 3, 23[2] D. Buchsbaum and D. Eisenbud, Algebraic structures for finite free resolutions, andsome structure theorems for ideals of codimension 3, American J. Math. (1977),447–485. 1, 2, 17[3] L. Bus´e, M. Chardin and A. Simis, Elimination and nonlinear equations of Rees alge-bras, J. Algebra (2010), 1314–1333. 3, 27[4] A. Conca and G. Valla, Betti numbers and liftings of Gorenstein codimension threeideals, Comm. Algebra (2000), 1371–1386. 2, 4295] B. Costa and A. Simis, Cremona maps defined by monomials, J. Pure Appl. Algebra, (2012), 212–225. 3, 24[6] S. J. Diesel, Some irreducibility and dimension theorems for families of height 3 Goren-stein algebras, Pacific J. Math. (1996), 365–397. 1, 2, 4[7] A. Doria and A. Simis, The Newton complementary dual revisited, J. Algebra and itsApplications, (2018), 1850004–1–16. 3, 24[8] J. Elias, Singular library for computing Macaulay’s inverse systems,arXiv:1501.01786v1 [math.AC] 8 Jan 2015. 4[9] J. Elias and M. E. Rossi, The structure of the inverse system of Gorenstein k -algebras,Adv. in Math. (2017), 306–327. 1, 4[10] M. Garrousian, A. Simis, S. O. Tohaneanu, A blowup algebra for hyperplane arrange-ments, Algebra Number Theory (2018), 1401–1429 22[11] T. Harima, A note on Artinian Gorenstein algebras of codimension three, J. Pure Appl.Algebra (1999), 45–56. 1[12] J. Herzog, A. Simis and W. V. Vasconcelos, Koszul homology and blowing-up rings,in Proc. Trento Conf. in Comm. Algebra, Lect. Notes in Pure and Applied Math. ,Marcel Dekker, New York, 1983, 79–169. 23[13] J. Herzog and N. V. Trung, Gr¨obner Bases and multiplicity of determinantal andpfaffian ideals, Adv. Math. (1992) 1–37. 19[14] J. Hong, A. Simis and W. V. Vasconcelos, On the equations of almost complete inter-sections, Bull. Braz. Math. Soc. (2012), 171-199. 1, 21[15] J. Hong, A. Simis and W. V. Vasconcelos, Ideals generated by quadrics, J. Algebra (2015), 177–189. 1, 2, 3, 17[16] C. Huneke and M. Miller, A note on the multiplicity of Cohen-Macaulay algebras withpure resolutions, Canad. J. Math. (1985), 1149–1162 23[17] A. Iarrobino, Associated graded algebra of a Gorenstein Artin algebra , Mem. Amer.Math. Soc. (1994), No. 514, Amer. Math. Soc. Providence. 1, 20[18] A. Iarrobino and V. Kanev,
Power Sums, Gorenstein Algebras, and DeterminantalVarieties . Appendix by A. Iarrobino and Steven L. Kleiman
The Gotzmann Theoremsand the Hilbert scheme , Lecture Notes in Mathematics, 1721. Springer-Verlag, Berlin,1999. xxxii+345 pp. ISBN: 3-540-66766-0. 1[19] A. Iarrobino and H. Srinivasan, Some Gorenstein Artin algebras of embedding dimen-sion four: components of
P GOR ( H ) for H = (1; 4; 7; :::; 1), J. of Pure and AppliedAlgebra (2005), 62–96. 1[20] J. Jelisiejew, Classifying local Artinian Gorenstein algebras, Collect. Math. (2017),101–127. 1 3021] S. El Khouri and A. Kustin, Artinian Gorenstein algebras with linear resolutions, J.Algebra (2014), 402–474. 1, 3, 4, 17, 18, 27[22] J.O. Kleppe, Maximal families of Gorenstein algebras, Trans. Amer. Math. Soc. (2006), 3133–3167. 1[23] A. R. Kustin, C. Polini and B. Ulrich, The equations defining blowup algebras of heightthree Gorenstein ideals, Algebra and Number theory, (2017), 1489–1525. 1, 3, 20[24] A. Kustin and B. Ulrich, If the socle fits, J. Algebra (1992), 63–80. 1, 10, 16[25] A. Ragusa and G. Zappal`a, Properties of 3-codimensional Gorenstein schemes, Comm.Algebra (2001), 303–318. 1[26] L. Reid, L. G. Roberts and M. Roitman, On complete intersections and their Hilbertfunctions, Canad. Math. Bull. (1991) 525–535. 3, 27[27] A. Simis, B. Ulrich and W. V. Vasconcelos, Codimension, multiplicities and integralextensions, Math. Proc. Cambridge Philos. Soc. (2001) 237–257 21, 23[28] W. V. Vasconcelos, Arithmetic of Blowup Algebras , London Math. Soc., Lecture NoteSeries 195, Cambridge University Press, Cambridge, 1994. 22[29] J. Watanabe, A note on Gorenstein rings of embedding dimension three, Nagoya Math.J. (1973), 227–232. 1 Addresses:
Dayane Lira
Departamento de Matem´aticaUniversidade Federal da Para´ıba58051-900 J. Pessoa, PB, Brazil e-mail : [email protected]
Zaqueu Ramos
Departamento de Matem´atica, CCETUniversidade Federal de Sergipe49100-000 S˜ao Cristov˜ao, Sergipe, Brazil e-mail : [email protected]
Aron Simis
Departamento de Matem´atica, CCENUniversidade Federal de Pernambuco50740-560 Recife, PE, Brazil e-maile-mail