G-biliaison of ladder Pfaffian varieties
GG-BILIAISON OF LADDER PFAFFIAN VARIETIES
E. DE NEGRI AND E. GORLA
Abstract:
The ideals generated by pfaffians of mixed size contained in a sublad-der of a skew-symmetric matrix of indeterminates define arithmetically Cohen-Macaulay,projectively normal, reduced and irreducible projective varieties. We show that thesevarieties belong to the G-biliaison class of a complete intersection. In particular, they areglicci.
Introduction
Pfaffian ideals and the varieties that they define have been studied both from thealgebraic and from the geometric point of view. In [1] Avramov showed that the idealsgenerated by pfaffians of fixed size define reduced and irreducible, projectively normalschemes. In this article, we study the ideals generated by pfaffians of mixed size containedin a subladder of a skew-symmetric matrix of indeterminates. Ideals generated by pfaffiansof the same size contained in a subladder of a skew-symmetric matrix of indeterminateswere already studied by the first author. In [7] it is shown that they define irreducibleprojective varieties, which are arithmetically Cohen-Macaulay and projectively normal.A necessary and sufficient condition for these schemes to be arithmetically Gorensteinis given in terms of the vertices of the defining ladder. In [6], one-cogenerated ideals ofpfaffians are studied. The deformation properties of schemes defined by pfaffians of fixedsize of a skew-symmetric matrix are studied in [13] and [14].In this paper, we study ladder ideals of pfaffians of mixed size from the point ofview of liaison theory (see [16] for an introduction to the subject, definitions and mainresults). A central open question in liaison theory asks whether every arithmeticallyCohen-Macaulay projective scheme is glicci (i.e. whether it belongs to the G-liaison classof a complete intersection of the same codimension). Migliore and Nagel have shown thatthe question has an affirmative answer up to deformation (see [17]). The main result ofthis paper is that ladder pfaffian varieties belong to the G-biliaison class of a linear variety.In particular they are glicci. The result is a natural extension to ideals of pfaffians of theresults established by the second author in [9], [10], and [11] for ideals of minors.
Mathematics Subject Classification.
Key words and phrases.
G-biliaison, Gorenstein liaison, pfaffian, ladder, complete intersection, arith-metically Gorenstein scheme, arithmetically Cohen-Macaulay scheme.The second author was partially supported by the Swiss National Science Foundation (grant no.107887) and by the Max-Planck-Institut f¨ur Mathematik, Bonn. She also wishes to express her gratitudeto the Mathematics Department of the University of Genova, where part of the work was done. a r X i v : . [ m a t h . AG ] S e p E. DE NEGRI AND E. GORLA
In the first section we study the ideals generated by pfaffians of mixed size containedin a subladder of a skew-symmetric matrix of indeterminates. We prove that they de-fine reduced and irreducible projective schemes (see Proposition 1.9), that we call ladderdeterminantal varieties. These varieties are shown to be arithmetically Cohen-Macaulayand projectively normal in Proposition 1.9. A localization argument is crucial for ex-tending these properties from the case of fixed size pfaffians of ladders to the case whenthe pfaffians have mixed size (see Proposition 1.8). In Proposition 1.10 we compute thecodimension of ladder pfaffian varieties.Section 2 contains the liaison results. In Theorem 2.3 we prove that ladder pfaffianvarieties belong to the G-biliaison class of a linear variety. Using standard liaison results,we conclude in Corollary 2.4 that they are (evenly) G-linked to a complete intersection.1.
Pfaffian ideals of ladders and ladder pfaffian varieties
Let V be a variety in P r = P rK , where K is an algebraically closed field of arbi-trary characteristic. Let I V be the saturated homogeneous ideal associated to V in thecoordinate ring of P r . Let I V ⊂ O P r be the ideal sheaf of V . Let W be a scheme thatcontains V . We denote by I V | W the ideal sheaf of V restricted to W , i.e. the quotientsheaf I V / I W .Let X = ( x ij ) be an n × n skew-symmetric matrix of indeterminates. In other words,the entries x ij with i < j are indeterminates, x ij = − x ji for i > j , and x ii = 0 for all i = 1 , ..., n . Let K [ X ] = K [ x ij | ≤ i < j ≤ n ] be the polynomial ring associated to X . Given a nonempty subset U = { u , ..., u p } of { , ..., n } we denote by [ u , · · · , u p ] the pfaffian of the matrix ( x ij ) i ∈U ,j ∈U . Definition 1.1. A ladder Y of X is a subset of the set { ( i, j ) ∈ N | ≤ i, j ≤ n } withthe following properties :(1) if ( i, j ) ∈ Y then ( j, i ) ∈ Y ,(2) if i < h, j > k and ( i, j ) , ( h, k ) belong to Y , then also ( i, k ) , ( i, h ) , ( h, j ) , ( j, k )belong to Y .We do not assume that a ladder Y is connected, nor that X is the smallest skew-symmetric matrix having Y as ladder.It is easy to see that any ladder can be decomposed as a union of square subladders(1) Y = X ∪ . . . ∪ X s where X k = { ( i, j ) | a k ≤ i, j ≤ b k } , for some integers 1 ≤ a ≤ . . . ≤ a s ≤ n and 1 ≤ b ≤ . . . ≤ b s ≤ n such that a k < b k forall k . We say that Y is the ladder with upper corners ( a , b ) , . . . , ( a s , b s ), and that X k isthe square subladder of Y with upper outside corner ( a k , b k ). We allow two upper cornersto have the same first or second coordinate, however we assume that no two upper cornerscoincide. Notice that with this convention a ladder does not have a unique decompositionof the form (1). In other words, a ladder does not correspond uniquely to a set of upper -BILIAISON OF LADDER PFAFFIAN VARIETIES 3 (a k ,b k ) . (b k ,a k ) . Figure 1.
A Laddercorners ( a , b ) , . . . , ( a s , b s ). However, a ladder is determined by its upper corners as in(1). Moreover, the upper corners of a ladder Y determine both the subladders X k and thesmallest skew-symmetric submatrix of X that has Y as ladder.Given a ladder Y we set Y = { x ij ∈ X | ( i, j ) ∈ Y , i < j } , and denote by K [ Y ] thepolynomial ring K [ x ij | x ij ∈ Y ]. If p is a positive integer, we let I p ( Y ) denote the idealgenerated by the set of the 2 p -pfaffians of X which involve only indeterminates of Y . Inparticular I p ( X ) is the ideal of K [ X ] generated by the 2 p -pfaffians of X .Whenever we consider a ladder Y , we assume that it comes with its set of uppercorners and the corresponding decomposition as a union of square subladders as in (1).Notice that the set of upper corners as given in our definition contains all the usualupper outside corners, and may contain some of the usual upper inside corners, as well asother elements of the ladder which are not corners of the ladder in the usual sense. Example 1.2.
Consider the set of upper corners { (1 , , (1 , , (3 , , (3 , , (4 , } . Then k = 5 and X = (1 ,
1) (1 , ,
1) (2 , X = (1 ,
1) (1 ,
2) (1 ,
3) (1 , ,
1) (2 ,
2) (2 ,
3) (2 , ,
1) (3 ,
2) (3 ,
3) (3 , ,
1) (4 ,
2) (4 ,
3) (4 , X = (3 ,
3) (3 , ,
3) (4 , X = (3 ,
3) (3 ,
4) (3 ,
5) (3 , ,
3) (4 ,
4) (4 ,
5) (4 , ,
3) (5 ,
4) (5 ,
5) (5 , ,
3) (6 ,
4) (6 ,
5) (6 , X = (4 ,
4) (4 ,
5) (4 ,
6) (4 , ,
4) (5 ,
5) (5 ,
6) (5 , ,
4) (6 ,
5) (6 ,
6) (6 , ,
4) (7 ,
5) (7 ,
6) (7 , E. DE NEGRI AND E. GORLA
The ladder determined by this choice of upper corners is Y = X ∪ X ∪ X ∪ X ∪ X = (1 ,
1) (1 ,
2) (1 ,
3) (1 , ,
1) (2 ,
2) (2 ,
3) (2 , ,
1) (3 ,
2) (3 ,
3) (3 ,
4) (3 ,
5) (3 , ,
1) (4 ,
2) (4 ,
3) (4 ,
4) (4 ,
5) (4 ,
6) (4 , ,
3) (5 ,
4) (5 ,
5) (5 ,
6) (5 , ,
3) (6 ,
4) (6 ,
5) (6 ,
6) (6 , ,
4) (7 ,
5) (7 ,
6) (7 , ,
2) is the upper outside corner of X , (1 ,
4) is the upper outside corner of X , (3 ,
4) isthe upper outside corner of X , (3 ,
6) is the upper outside corner of X , and (4 ,
7) is theupper outside corner of X .Notice that our set of upper corners contains (3 , , ,
2) which is not a cornerin the usual terminology. It contains also all the usual upper outside corners, namely(1 , , (3 , , and (4 , X = x , x , x , x , x , x , − x , x , x , x , x , x , − x , − x , x , x , x , x , − x , − x , − x , x , x , x , − x , − x , − x , − x , x , x , − x , − x , − x , − x , − x , x , − x , − x , − x , − x , − x , − x , be the smallest skew-symmetric matrix having Y as ladder. The set of indeterminatescorresponding to Y is Y = x , x , x , x , x , x , x , x , x , x , x , x , x , x , Definition 1.3.
Let Y = X ∪ . . . ∪ X s be a ladder as in Definition 1.1.Let X k = { x i,j | ( i, j ) ∈ X k , i < j } for k = 1 , . . . , s . Fix a vector t = ( t , . . . , t s ), t ∈ { , . . . , (cid:98) n (cid:99)} s . The pfaffian ideal I t ( Y ) is by definition the sum of pfaffian ideals I t ( X ) + . . . + I t s ( X s ) ⊆ K [ Y ]. Sometimes we refer to these ideals as pfaffian ideals ofladders . Example 1.4.
Let Y be the ladder of Example 1.2, together with the same choice ofupper corners. Let t = (1 , , , , I t ( Y ) = ( x , , − x , x , + x , x , , x , , − x , x , + x , x , , x , x , − x , x , + x , x , ) ⊆ K [ x , , x , , x , , x , , x , , x , , x , , x , , x , , x , , x , , x , , x , , x , ] .I t ( Y ) is the saturated ideal of a variety of codimension 5 in P . -BILIAISON OF LADDER PFAFFIAN VARIETIES 5 Remarks 1.5. (1) Let
Z ⊇ Y be two ladders of X , and let Z, Y be the correspondingsets of indeterminates. We have an isomorphism of graded K -algebras K [ Y ] /I t ( Y ) ∼ = K [ Z ] /I t ( Y ) + ( x ij | x ij ∈ Z \ Y ) ∼ = K [ X ] /I t ( Y ) + ( x ij | x ij ∈ X \ Y ) . Here I t ( Y ) is an ideal in K [ Y ] , K [ X ] , and K [ Z ] respectively. Then the height ofthe ideal I t ( Y ) does not depend of whether we regard it as an ideal of K [ Y ] , K [ X ] , or K [ Z ].(2) We can assume without loss of generality that2 t k ≤ b k − a k + 1 . In fact, if 2 t k > b k − a k + 1 then I t k ( X k ) = 0.(3) Moreover, we can assume that a k − a k − > t k − − t k and b k − b k − > t k − t k − , for k = 2 , . . . , s .If a k − a k − ≤ t k − − t k , by successively developing each 2 t k − -pfaffian of X k − withrespect to the first 2( a k − a k − ) rows and columns, we obtain an expression of thepfaffian as a combination of pfaffians of size 2 t k − − a k − a k − ) ≥ t k that involveonly rows and columns from X k . Therefore I t k ( X k ) ⊇ I t k − ( X k − ). Similarly, if b k − b k − ≤ t k − t k − , by developing each 2 t k -pfaffian of X k with respect to thelast 2( b k − b k − ) rows and columns, we obtain an expression of the pfaffian as acombination of pfaffians of size 2 t k − b k − b k − ) ≥ t k − that involve only rowsand columns from X k − . Therefore I t k ( X k ) ⊆ I t k − ( X k − ). In either case, we canremove a part of the ladder and reduce to the study of a proper subladder thatcorresponds to the same pfaffian ideal. For example, if b k − b k − ≤ t k − t k − wecan consider the ladder˜ Y = X ∪ . . . ∪ X k − ∪ X k +1 ∪ . . . ∪ X s and let t (cid:48) = ( t , . . . , t k − , t k +1 , . . . , t s ) . Since I t k ( X k ) ⊆ I t k − ( X k − ), we have I t ( Y ) = I t (cid:48) ( ˜ Y ) , where ˜ Y is the set ofindeterminates corresponding to ˜ Y .The class of pfaffian ideals that we consider is very large. We now give examples ofinteresting families of ideals generated by pfaffians, which belong to the class of pfaffianideals that we study. Examples 1.6. (1) If t = ( t, . . . , t ) ∈ { , . . . , (cid:98) n (cid:99)} s then I t ( Y ) is the ideal generatedby the pfaffians of size 2 t of X that involve only indeterminates from Y . In [7] itis proven that in this case K [ Y ] /I t ( Y ) is a Cohen-Macaulay normal domain.(2) If we choose all the upper corners on the same row, we obtain ideals of pfaffians ofmatrices which are contained one in the other. In this case 1 = a = a = . . . = a s ,hence 1 < b < b < . . . < b s = n . By Remark 1.5 (3), this forces 1 ≤ t < t <. . . < t s . The ideal I t ( X ) is generated by the 2 t i -pfaffians of the submatrix of thefirst b i rows and columns, i = 1 , ..., s .Similarly, if we choose all the upper corners on the same column, that is, 1 = a < a < . . . < a s and b = b = . . . = b s = n , we obtain the ideal generatedby the 2 t i -pfaffians of the last n − a i + 1 rows and columns, i = 1 , ..., s , with E. DE NEGRI AND E. GORLA t > t > . . . > t s ≥
1. Notice that these two choices of upper corners produce thesame family of ideals.(3) Consider the ladder with two upper corners (1 , b ) , (1 , n ), b < n , and the vector( t , t ) = ( t, t + 1). Then the ideal I t ( Y ) is generated by all the 2 t + 2-pfaffiansand by the 2 t -pfaffians of the first b rows and columns, of an n × n skew-symmetricmatrix of indeterminates. This ideals belong to a well-known class of ideals gen-erated by pfaffians in a matrix, the cogenerated ideals, which have been studiedin [6]. In fact I t ( Y ) is the ideal cogenerated by the pfaffian [1 , , . . . , t − , b + 1].Notice however that not every one-cogenerated pfaffian ideal is a pfaffian ideal ofladders.We will show that for every vector t the ideals I t ( Y ) are prime (see Proposition 1.9).Therefore they define reduced and irreducible projective varieties. Definition 1.7.
Let V ⊆ P r . V is a pfaffian variety if I V = I t ( X ), where X is askew symmetric matrix of indeterminates of size n × n . V is a ladder pfaffian variety if I V = I t ( Y ) = I t ( X ) + . . . + I t s ( X s ) for some ladder Y = X ∪ . . . ∪ X s and some vector t = ( t , . . . , t s ) ∈ { , . . . , (cid:98) n (cid:99)} s .Notice that every pfaffian variety is a ladder pfaffian variety. Therefore, from nowon we will only consider ladder pfaffian varieties. Moreover, in view of Remark 1.5 (1) wewill not distinguish between ladder pfaffian varieties and cones over them.In this section we study ladder pfaffian varieties. We prove that their saturated idealsare generated by pfaffians of mixed size contained in a subladder of a skew-symmetric ma-trix of indeterminates, or in other words that the ideals in question are prime. We provethat the ladder pfaffian varieties are arithmetically Cohen-Macaulay and projectively nor-mal, and we compute their codimension. We choose to follow a classical commutativealgebra localization argument to approach the problem. Some of our results could be ob-tained also by using a Schubert calculus approach, at least for the case of ideals of pfaffiansof fixed size in a matrix, which define Schubert varieties in orthogonal Grassmannians.In order to establish the properties that we just mentioned, we will make use of alocalization argument (analogous to that of Lemma 7.3.3 in [2]). The following propositionwill be crucial in the sequel. We use the notation of Definition 1.1 and Definition 1.3, andrefer to Fig. 2. Notice that if t l ≥ l , then it is always possible to choose a k such t k ≥ a k +1 − ≥ a k , and b k ≥ b k − + 1. In fact, it suffices to choose k such that t k ≥ t l for all l , and the inequalities follow from Remark 1.5 (3). Notice moreover thatfor a classical ladder (i.e. a ladder for which no two vertices belong to the same row orcolumn) these conditions are automatically satisfied. Proposition 1.8.
Let Y = X ∪ · · · ∪ X s be a ladder of a skew-symmetric matrix X ofindeterminates. Let t = ( t , . . . , t s ) ∈ { , . . . , (cid:98) n (cid:99)} s , and let I t ( Y ) be the correspondingpfaffian ideal. Fix k ∈ { , . . . , s } such that t k ≥ , a k +1 − ≥ a k , and b k ≥ b k − + 1 . Let t (cid:48) = ( t , . . . , t k − , t k − , t k +1 , . . . , t s ) . Let Y (cid:48) be the subladder of Y with outside corners ( a , b ) , . . . , ( a k − , b k − ) , ( a k + 1 , b k − , ( a k +1 , b k +1 ) , . . . , ( a s , b s ) . -BILIAISON OF LADDER PFAFFIAN VARIETIES 7 (a k ,b k ) (a k +1,b k -1 ) .. Figure 2.
Then there is an isomorphism K [ Y ] /I t ( Y )[ x − a k ,b k ] ∼ = K [ Y (cid:48) ] /I t (cid:48) ( Y (cid:48) )[ x a k ,b k − +1 , . . . , x a k ,b k , x a k +1 ,b k , . . . , x a k +1 − ,b k ][ x − a k ,b k ] . Proof.
We prove the proposition in the case s = k = 2. In the general case the proofworks exactly the same way. Let Y = X ∪ X be the ladder with upper corners (1 , c )and ( a, b ), t = ( t , t ) with t ≥
2. Let X = ( x i,j ) ≤ i,j ≤ c and X = ( x i,j ) a ≤ i,j ≤ b be twosubmatrices of X . Let ˜ X = (˜ x u,v ) be a skew-symmetric matrix of indeterminates of size c × c , whose entries have indexes 1 ≤ u, v ≤ c , where ˜ x u,v = x u,v if u < a or v < a , and˜ x u,v are new indeterminates whenever a ≤ u < v ≤ c (that is ( u, v ) ∈ X ).Let ˜ Y be the set of all the indeterminates in the matrices ˜ X and X , and denoteby L the ideal ( { x u,v − ˜ x u,v } a ≤ u Using Proposition 1.8 we can establish some properties of ladder pfaffian varieties. Proposition 1.9. Pfaffian ideals of ladders define reduced and irreducible, arithmeticallyCohen-Macaulay projectively normal varieties.Proof. Let I t ( Y ) be a pfaffian ideal. Let t max be the maximum of { t , ..., t s } . If t max = 1then I t ( X ) is generated by indeterminates, und we are done. Assume that t max ≥ (cid:98) Y be the ladder obtained by enlarging Y along its borders by the region whichincreases the size of every X k by t max − t k . Thus (cid:98) Y is the ladder with upper corners( a k − t max + t k , b k + t max − t k ), with k = 1 , . . . , s . Let (cid:98) Y = { x ij ∈ X | ( i, j ) ∈ (cid:98) Y , i < j } and let Ψ = (cid:98) Y \ Y . By Proposition 1.8, we can repeatedly localize K [ (cid:98) Y ] /I t max ( (cid:98) Y ) atappropriate upper outside corners and obtain the original ladder Y and the pfaffians ofsize t , . . . , t s . It follows that there exists a subset { z , ..., z p } of Ψ such that K [ (cid:98) Y ] /I t max ( (cid:98) Y )[ z − , . . . , z − p ] ∼ = K [ Y ] /I t ( Y )[Ψ][ z − , . . . , z − p ] . By [7, 1.2,2.1,3.5] one has that K [ (cid:98) Y ] /I t max ( (cid:98) Y ) is a Cohen-Macaulay normal domain,thus K [ Y ] /I t ( Y )[Ψ][ u − , . . . , u − q ] is a Cohen-Macaulay normal domain. Since Ψ is aset of indeterminates over K [ Y ] /I t ( Y ) and { u , ...., u q } ⊂ Ψ, then also K [ Y ] /I t ( Y )[Ψ]is a Cohen-Macaulay normal domain. Hence I t ( Y ) defines a reduced and irreducible,arithmetically Cohen-Macaulay normal projective variety. (cid:3) A standard argument allows us to compute the codimension of ladder pfaffian vari-eties. The notation is the same as in Proposition 1.8. Proposition 1.10. Let Y be a ladder with upper corners ( a , b ) , . . . , ( a s , b s ) . Let L = { ( i, j ) | a k + t k − ≤ i, j ≤ b k − t k + 1 , for some ≤ k ≤ s } be a subset of Y . Then L is a ladder and the height of I t ( Y ) is equal to the cardinalityof { ( i, j ) ∈ L | i < j } .Proof. Observe that a k + t k − > a k − + t k − − , and b k − t k + 1 > b k − − t k − + 1by Remark 1.5 (3). Moreover by Remark 1.5 (2) we have b k − a k > t k − 2. Then b k − t k + 1 > a k + t k − k . Therefore L is a ladder with upper corners { ( a k + t k − , b k − t k +1) | k = 1 , . . . , s } . Notice that L has no two corners on the same row or column. Let L = { x i,j | ( i, j ) ∈L , i < j } .We argue by induction on τ = t + . . . + t s ≥ s . If τ = s , then t = . . . = t s = 1,and L = Y . Moreover, I ( Y ) = ( x ij | x ij ∈ Y, i < j ) = ( x ij | x ij ∈ L, i < j ) , thus the thesis holds true. -BILIAISON OF LADDER PFAFFIAN VARIETIES 9 Assume then that the thesis is true for τ − ≥ s and prove it for τ . Since τ > s ,then t k ≥ k . By Proposition 1.8 we have an isomorphism K [ Y ] /I t ( Y )[ x − a k ,b k ] ∼ = K [ Y (cid:48) ] /I t (cid:48) ( Y (cid:48) )[ x a k ,b k − +1 , . . . , x a k ,b k , x a k ,b k +1 , . . . , x a k ,b k +1 − ][ x − a k ,b k ] . Since x a k ,b k does not divide zero modulo I t (cid:48) ( Y (cid:48) ) and I t ( Y ), we haveht I t ( Y ) = ht I t (cid:48) ( Y (cid:48) ) . Notice that the same ladder L computes the height of both I t (cid:48) ( Y (cid:48) ) and I t ( Y ), thus thethesis follows by the induction hypothesis. (cid:3) Linkage of ladder pfaffian varieties In this section we prove that ladder pfaffian varieties belong to the G-biliaison class ofa complete intersection. The biliaisons are performed on ladder pfaffian varieties, whichare reduced and irreducible (hence generically Gorenstein), and arithmetically Cohen-Macaulay. Therefore we can conclude that ladder pfaffian varieties are glicci. Noticethe analogy with determinantal varieties, symmetric determinantal varieties and mixedladder determinantal varieties, that were treated by the second author with analogoustechniques in [9], [10], and [11].The following lemma due to De Concini and Procesi [5, 6.1] will be needed in thesequel. Lemma 2.1. Let A be a skew symmetric n × n matrix, p, m ≤ n even integers and c , ..., c p , d , ..., d m elements of the set { , ..., n } . Then [ c , ..., c p ][ d , ..., d m ] − p (cid:88) h =1 [ c , ..., c h − , d , c h +1 , ..., c p ][ c h , d , ..., d m ] = m (cid:88) k =2 ( − k − [ d k , d , c , ..., c p ][ d , ..., d k − , d k +1 , ..., d m ] where [ ... ] denotes a pfaffian of A . The following result will also be needed in the proof. We will use it to constructthe ladder pfaffian varieties on which we perform the G-biliaisons. We follow the notationestablished in Definitions 1.1 and 1.3. Lemma 2.2. Let V ⊆ P r be a ladder pfaffian variety of codimension c . Let Y be theladder corresponding to V , and assume that t k = max { t , . . . , t s } ≥ . Let Z be thesubladder of Y with upper corners ( a , b ) , . . . , ( a k − , b k − ) , ( a k , b k − , ( a k + 1 , b k ) , ( a k +1 , b k +1 ) , . . . , ( a s , b s ) and let u = ( t , . . . , t k − , t k , t k , t k +1 , . . . , t s ) . Then the ladder pfaffian variety W ⊆ P r with I W = I u ( Z ) has codimension c − . Proof. We decompose the ladder Z as Z = X ∪ . . . ∪ X k − ∪ X (1) k ∪ X (2) k ∪ X k +1 ∪ . . . ∪ X s where X (1) k , X (2) k are the square subladders with upper outside corner ( a k , b k − 1) and( a k + 1 , b k ), respectively. Let u = ( t , . . . , t k − , t k , t k , t k +1 , . . . , t s ) , u ∈ { , . . . , (cid:98) n (cid:99)} s +1 . If the ladder Z satisfies the inequalities of Remark 1.5 (2) and (3), then the codi-mension count follows from Proposition 1.10. In fact, the codimension c of V equals thecardinality of the subset { ( i, j ) ∈ L | i < j } where L is the ladder with upper corners( a + t − , b − t + 1) , . . . , ( a s + t s − , b s − t s + 1) . The codimension of W equalscardinality of { ( i, j ) ∈ L (cid:48) | i < j } , where L (cid:48) is the ladder obtained from L by removing( a k + t k − , b k − t k +1) and ( b k − t k +1 , a k + t k − W has codimension c − Z may not satisfy the inequalities of Remark 1.5(2),(3) even under the assumption that the ladder Y does. In particular, the followingthree situations may occur:(1) 2 t k = b k − a k + 1 > b k − a k = ( b k − − a k + 1 = b k − ( a k + 1) + 1,(2) a k +1 − ( a k + 1) = t k − t k +1 ,(3) ( b k − − b k − = t k − t k − .In case (1) we delete the subladders X (1) k and X (2) k , and let Z = X ∪ . . . ∪ X k − ∪ X k +1 ∪ . . . ∪ X s and u = ( t , . . . , t k − , t k +1 , . . . , t s ) . In case (2) we delete the subladder X (2) k , and let Z = X ∪ . . . ∪ X k − ∪ X (1) k ∪ X k +1 ∪ . . . ∪ X s and u = ( t , . . . , t k − , t k , t k +1 , . . . , t s ) . In case (3) we delete the subladder X (1) k , and let Z = X ∪ . . . ∪ X k − ∪ X (2) k ∪ X k +1 ∪ . . . ∪ X s and u = ( t , . . . , t k − , t k , t k +1 , . . . , t s ) . Notice that it may happen that more than one of the cases (1), (2) and (3) is verifiedfor the ladder Z . In this case, we behave as if we were in the situation (1). As we alreadyobserved, none of the operations above affects the ideal I W .If we are in situation (1), then 2 t k = b k − a k + 1. Applying Proposition 1.10 to theladder Z = X ∪ . . . ∪ X k − ∪ X k +1 ∪ . . . ∪ X s we obtain that the codimension of W equalsthe cardinality of { ( i, j ) ∈ L (cid:48) | i < j } , where L (cid:48) is the ladder with upper corners( a + t − , b − t + 1) , . . . , ( a k − + t k − − , b k − − t k − + 1) , ( a k +1 + t k +1 − , b k +1 − t k +1 + 1) , . . . , ( a s + t s − , b s − t s + 1) . -BILIAISON OF LADDER PFAFFIAN VARIETIES 11 Since a k + t k − b k − t k and a k + t k = b k − t k + 1, the cardinality of { ( i, j ) ∈ L (cid:48) | i < j } coincides with the cardinality of { ( i, j ) ∈ L (cid:48)(cid:48) | i < j } , where L (cid:48)(cid:48) has upper corners( a + t − , b − t + 1) , . . . , ( a k − + t k − − , b k − − t k − + 1) , ( a k + t k − , b k − t k ) , ( a k + t k , b k − t k + 1) , ( a k +1 + t k +1 − , b k +1 − t k +1 + 1) , . . . , ( a s + t s − , b s − t s + 1) . Equivalently, L (cid:48)(cid:48) is obtained from L by removing ( a k + t k − , b k − t k + 1) and its symmetricpoint ( b k − t k + 1 , a k + t k − W is c − Z of case (1). The codimension of W equals the cardinality { ( i, j ) ∈ L (cid:48) | i < j } , where L (cid:48) is the same as in case (1). Since a k + t k = a k +1 + t k +1 − b k − t k = b k − − t k − + 1, the cardinality of { ( i, j ) ∈ L (cid:48) | i < j } coincides with thecardinality of { ( i, j ) ∈ L (cid:48)(cid:48) | i < j } , where L (cid:48)(cid:48) is the same as in case (1). In fact, the angles( a k + t k , b k − t k + 1) and ( a k +1 + t k +1 − , b k +1 − t k +1 + 1) are on the same row. Moreoverthe angles ( a k − + t k − − , b k − − t k − + 1) and ( a k + t k − , b k − t k ) are on the samecolumn. We conclude that the codimension of W is c − a k +1 − ( a k +1) = t k − t k +1 . Assume that ( b k − − b k − >t k − t k − . Apply Proposition 1.10 to the ladder Z = X ∪ . . . ∪ X k − ∪ X (1) k ∪ X k +1 ∪ . . . ∪ X s .The codimension of W equals the cardinality of { ( i, j ) ∈ L (cid:48) | i < j } , where L (cid:48) is the ladderwith upper corners( a + t − , b − t + 1) , . . . , ( a k − + t k − − , b k − − t k − + 1) , ( a k + t k − , b k − t k )( a k +1 + t k +1 − , b k +1 − t k +1 + 1) , . . . , ( a s + t s − , b s − t s + 1) . Since a k + t k = a k +1 + t k +1 − 1, the cardinality of { ( i, j ) ∈ L (cid:48) | i < j } coincides withthe cardinality of { ( i, j ) ∈ L (cid:48)(cid:48) | i < j } , where L (cid:48)(cid:48) is the same as in case (1). In fact, theangles ( a k + t k , b k − t k + 1) and ( a k +1 + t k +1 − , b k +1 − t k +1 + 1) are on the same row. Weconclude that the codimension of W is c − Z = X ∪ . . . ∪X k − ∪ X (2) k ∪ X k +1 ∪ . . . ∪ X s and observe that X (1) k can be disregarded in the codimensioncount, as the angles ( a k − + t k − − , b k − − t k − + 1) and ( a k + t k − , b k − t k ) are on thesame column. (cid:3) The next theorem is the main result of this paper. The idea of the proof is as follows:starting from a ladder pfaffian variety V , we construct two ladder pfaffian varieties V (cid:48) and W such that V and V (cid:48) are generalized divisors on W . Then we show how V (cid:48) can beobtained from V by an elementary G-biliaison on W . Theorem 2.3. Any ladder pfaffian variety can be obtained from a linear variety by afinite sequence of ascending elementary G-biliaisons.Proof. Let V be a ladder pfaffian variety. Let Y be the ladder corresponding to V , I V = I t ( Y ) = I t ( X ) + · · · + I t s ( X s ) ⊆ K [ Y ] . We perform all the linkage steps in P r = Proj ( K [ Y ]). If t = . . . = t s = 1 then V is alinear variety. Therefore we consider the case when t k = max { t , . . . , t s } ≥ 2. It follows (a k ,b k ) (a k +1,b k -1 ) .. Figure 3. that a k +1 − a k > t k − t k +1 ≥ b k − b k − > t k − t k − ≥ 0, therefore a k − < a k + 1 ≤ a k +1 and b k − ≤ b k − < b k +1 . Let Y (cid:48) be the ladder with upper corners( a , b ) , . . . , ( a k − , b k − ) , ( a k + 1 , b k − , ( a k +1 , b k +1 ) , . . . , ( a s , b s )and let t (cid:48) = ( t , . . . , t k − , t k − , t k +1 , . . . , t s ). Let Y (cid:48) = X ∪ . . . ∪ X k − ∪ X (cid:48) k ∪ X k +1 ∪ . . . ∪ X s , where X (cid:48) k is the square subladder with upper outside corner ( a k + 1 , b k − 1) (see Figure2). Notice that all the inequalities of Remark 1.5 (2),(3) are satisfied by Y (cid:48) , t (cid:48) . Let V (cid:48) bethe ladder pfaffian variety with saturated ideal I V (cid:48) = I t (cid:48) ( Y (cid:48) ) . By Proposition 1.10 andRemark 1.5 (1), V and V (cid:48) have the same codimension c in P r = Proj ( K [ Y ]). In fact, inboth cases c equals the cardinality of the subset { ( i, j ) ∈ L | i < j } where L is the ladderwith upper corners ( a + t − , b − t + 1) , . . . , ( a s + t s − , b s − t s + 1) . Let Z be the ladder obtained from Y by removing ( a k , b k ) and ( b k , a k ) (see Figure3), and let u = ( t , . . . , t k − , t k , t k , t k +1 , . . . , t s ).Let W be the ladder pfaffian variety with I W = I u ( Z ). W has codimension c − P r = Proj ( K [ Y ]), by Lemma 2.2. Clearly I V ⊇ I W , therefore V ⊆ W . We claim thatalso V (cid:48) ⊆ W . In order to show that I t (cid:48) ( Y (cid:48) ) ⊇ I u ( Z ), it suffices to show that I t k ( X (1) k ) + I t k ( X (2) k ) ⊆ I t k − ( X (cid:48) k ). Let f = [ u , . . . , u t k ] be a 2 t k -pfaffian in I t k ( X (1) k ) + I t k ( X (2) k ),with a k ≤ u < u < · · · ≤ u t k ≤ b k . If a k , b k (cid:54)∈ { u , . . . , u t k } , then f ∈ I t k ( X (cid:48) k ) ⊆ I t k − ( X (cid:48) k ). If a k = u then b k (cid:54)∈ { u , . . . , u t k } , since f does not involve the indeterminate x a k ,b k . By expanding f along the u -th row and column one has f = t k (cid:88) h =1 ± [ u , u h ][ u , . . . , ˇ u h , . . . , u t k ] . The notation [ ..., ˇ u i , ... ] means that the index u i is not involved in the pfaffian. Since[ u , . . . , ˇ u h , . . . , u t k ] ∈ I t k − ( X (cid:48) k ), one has f ∈ I t k − ( X (cid:48) k ). Similarly, if u t k = b k then -BILIAISON OF LADDER PFAFFIAN VARIETIES 13 a k (cid:54)∈ { u , . . . , u t k − } , and expanding f along the u t k -th row and column the conclusionfollows.The variety W is reduced, irreducible, and arithmetically Cohen-Macaulay by Propo-sition 1.9. In particular it is generically Gorenstein. Therefore we can regard V and V (cid:48) as generalized divisors on W (see [12] about the theory of generalized divisors). Then V and V (cid:48) are G-bilinked on W if and only if V ∼ V (cid:48) + mH for some m ∈ Z , where H is thehyperplane section divisor on W . This is in turn equivalent to(2) I V | W ∼ = I V (cid:48) | W ( − m )as subsheaves of the sheaf of total quotient rings of W . In order to construct an isomor-phism as (2), we prove that[ a k , u , . . . , u t k − , b k ][ u , . . . , u t k − ] = [ a k , v , . . . , v t k − , b k ][ v , . . . , v t k − ]modulo I W , for any choice of u i , v i such that a k < u < . . . < u t k − < b k and a k Every ladder pfaffian variety V can be G-linked in t + . . . + t s − s ) steps to a linear variety of the same codimension. In particular, ladder pfaffian varietiesare glicci. Remark 2.5. The varieties cut out by pfaffians of fixed size of a skew-symmetric matrix(i.e. those for which Y = X and t = . . . = t s ) are known to be arithmetically Gorenstein(see [1] and [14]). In [7] ideals generated by pfaffians of fixed size in a ladder are considered,and a characterization is given of the ones defining arithmetically Gorenstein varieties. The results in [4] play a central role in the argument. It turns out that the arithmeticallyGorenstein varieties are essentially only those cut out by pfaffians of fixed size of a skew-symmetric matrix, and a few more cases that are directly connected to those. Notice thatcombining Proposition 1.8 and the results in [7], one easily obtains a characterizationof the arithmetically Gorenstein ladder pfaffian varieties for pfaffians of mixed size, interms of the upper outside corners of the ladder and of the vector t . The technique ofTheorem 6.3.1 in [8] applies to this situation.Arithmetically Gorenstein schemes are known to be glicci (see Theorem 7.1 of [3]).Notice however that only very special ladder pfaffian varieties are arithmetically Goren-stein. 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Nagel, Monomial ideals and the Gorenstein liaison class of a complete intersection,Compositio Math. (2002), no. 1, 25–36. -BILIAISON OF LADDER PFAFFIAN VARIETIES 15 Dipartimento di Matematica, Universit`a di Genova,via Dodecaneso 35, 16146 Genova, Italy E-mail address : [email protected] Institut f¨ur Mathematik, Universit¨at Z¨urich,Winterthurerstrasse 190, CH-8057 Z¨urich, Switzerland E-mail address ::