Rees algebras of sparse determinantal ideals
Ela Celikbas, Emilie Dufresne, Louiza Fouli, Elisa Gorla, Kuei-Nuan Lin, Claudia Polini, Irena Swanson
aa r X i v : . [ m a t h . A C ] J a n REES ALGEBRAS OF SPARSE DETERMINANTAL IDEALS
ELA CELIKBAS, EMILIE DUFRESNE, LOUIZA FOULI, ELISA GORLA, KUEI-NUAN LIN,CLAUDIA POLINI, AND IRENA SWANSON
Abstract.
We determine the defining equations of the Rees algebra and of the specialfiber ring of the ideal of maximal minors of a 2 × n sparse matrix. We prove that theirinitial algebras are ladder determinantal rings. This allows us to show that the Rees algebraand the special fiber ring are Cohen-Macaulay domains, they are Koszul, they have rationalsingularities in characteristic zero and are F-rational in positive characteristic. Introduction
Given an ideal I in a Noetherian ring R , one can associate an algebra to I known as theRees algebra R ( I ) of I . This algebra R ( I ) = L i ≥ I i t i is a subalgebra of R [ t ], where t isan indeterminate. It was introduced by Rees in 1956 in order to prove what is now knownas the Artin-Rees Lemma [34]. Geometrically, the Rees algebra corresponds to the blowupof Spec( R ) along V ( I ). If R is local with maximal ideal m or graded with homogeneousmaximal ideal m , the special fiber ring of I is the algebra F ( I ) = R ( I ) ⊗ R/ m . Thisalgebra corresponds to the special fiber of the blowup of Spec( R ) along V ( I ). Besides itsconnections to resolution of singularities, the study of Rees algebras plays an important rolein many other active areas of research including multiplicity theory, equisingularity theory,asymptotic properties of ideals, and integral dependence.Although blowing up is a fundamental operation in the study of birational varieties,an explicit understanding of this process remains an open problem. In particular, a keyobjective in this area is to express the Rees algebra and the special fiber ring as quotientsof a polynomial ring, henceforth to determine their defining ideals.This question is wide open even for the simplest classes of ideals, including ideals gen-erated by forms of the same degree in a polynomial ring. These are precisely the idealsgenerated by forms parametrizing a variety in projective space. The implicit equations ofthese varieties can be obtained from the defining ideal of the Rees ring. Indeed, the biho-mogenous coordinate ring of the graph of the morphism defined by the forms is the Reesalgebra of the ideal I . The homogeneous coordinate ring of the variety parametrized by theforms is the special fiber ring.As the graph of a map carries more information than its image, even a partial under-standing of the Rees ring such as the bigraded degrees of its defining equations, the Betti Key words and phrases.
Rees algebra, special fiber ring, determinantal ideal, sparse matrix, toric ring,Koszul algebra, ladder determinantal ring, SAGBI basis, Gr¨obner basis, Pl¨ucker relations.2020
Mathematics Subject Classification . Primary 13A30, 13C40; Secondary 14M12, 13P10, 05E40, 13F50.Claudia Polini was partially supported by NSF grant DMS-1902033. umbers, or the regularity of the defining ideal can be instrumental to the study of thevariety. Determining the defining equations of the Rees algebra is a difficult problem inelimination theory, studied by commutative algebraists, algebraic geometers, and appliedmathematicians in geometric modeling, see e.g. [7, 8, 9, 14, 37]. Answers to these questionsalso have applications to the study of chemical reaction networks [15].The goal of this paper is to determine the defining equations of the Rees algebra and ofthe special fiber ring of the ideals generated by the maximal minors of sparse 2 × n matrices.Sparse matrices are matrices whose entries are either zeroes or distinct variables. Theirdegeneracy loci were first studied by Giusti and Merle in the 80’s. In [23] they compute thecodimension of their defining ideals and characterize when these ideals are prime or Cohen-Macaulay. Boocher in 2012 proved in [2] that a minimal free resolution of the ideals ofmaximal minors of sparse matrices can be obtained from the Eagon-Northcott complex viaa pruning method. In the same paper, he shows that the natural generators form a universalGr¨obner basis.In the case of a generic matrix, that is a matrix whose entries are distinct variables, thespecial fiber ring of the ideal of maximal minors is the coordinate ring of a Grassmannianvariety. The fact that the Pl¨ucker relations define the Grassmannian variety is a classicaltheorem, see e.g. [6]. The Rees algebra and the special fiber ring of ideals of maximalminors of generic matrices are Algebras with Straightening Laws (ASLs) in the sense of [16],see [18, 19]. Since the straightening relations come from the Pl¨ucker relations and thedefining equations of the symmetric algebra, it follows that R ( I ) is of fiber type, see [16, 18],[5, Lemma 2.2.1], [6, Proposition 4.2].In addition, as the posets defining R ( I ) and F ( I ) are wonderful in the sense of [19], itfollows that both algebras are Cohen-Macaulay, see [19, Proposition 2.6]. The normalityof F ( I ) follows immediately from the Cohen-Macaulay property since the Grassmannianvariety is smooth. Trung in [40] proved that the powers and symbolic powers of I coincideand therefore R ( I ) is normal. From the deformation theorem developed in [12], see forinstance [4, Proposition 3.6], one can see that the Rees algebra of the ideal of maximalminors is defined by a Gr¨obner basis of quadrics. The same statement holds for F ( I ).Therefore, in the generic case both R ( I ) and F ( I ) are Koszul algebras and according to [3]the ideal I and all its powers have a linear resolution.Our main result shows that the above properties of the blowup algebras of ideals ofmaximal minors of a generic matrix still hold in the case of sparse 2 × n matrices.Now let I be the ideal of maximal minors of a 2 × n sparse matrix. Inspired by thepioneering work of Conca, Herzog, and Valla [12], we study the initial algebra of the Reesalgebra of I . Our main technique is SAGBI bases [30, 35], an analogue for algebras ofGr¨obner bases for ideals.This approach was successfully used to study the Rees algebras of other families ofideals [1, 12, 32].First we prove that the initial algebra of the Rees algebra of I is the Rees algebra ofthe initial ideal of I with respect to a suitable order (see Theorem 3.3). Using deformationtheory, we transfer properties from the Rees algebra of the initial ideal to the Rees algebraof the ideal itself. One advantage of this approach is that it allows us to reduce to the study f the Rees algebra of the initial ideal, which is not just a monomial algebra, but also theRees algebra of the edge ideal of a graph and a ladder determinantal ring. These objectshave been studied extensively and one can draw a plethora of information that allows us todescribe these algebras in full detail, see among others [10, 13, 36, 41, 42, 43].A key step in our proof that R (in τ ( I )) is the initial algebra of R ( I ) is Lemma 3.2, wherewe prove that taking the initial ideal commutes with powers. The main idea behind theproof is a comparison of the Hilbert functions of I and (in τ ( I )) , an approach which wasfirst used in [25]. We then use a lifting technique to obtain the defining equations of the Reesalgebra and of the special fiber ring. Interestingly, they turn out to be the specialization ofthe defining equations of the Rees algebra and of the special fiber ring in the generic case.The general question of understanding the Rees algebra and the special fiber ring of theideal I of maximal minors of a sparse m × n matrix is still open. In Remark 4.11 we proposea different approach, which applies to sparse matrices whose zero region has a special shape.These sparse matrices are exactly those that have the property that a maximal minor is non-zero if and only if the product of the elements on its diagonal is non-zero. This yields a nicecombinatorial description of the initial ideal of I . Our arguments allow us to compute theequations of the special fiber ring and the Rees algebra and to establish algebraic propertiessuch as normality, Cohen-Macaulayness, and Koszulness. We conjecture that these propertieshold in general.Our main results are summarized in the following. Theorem 1.1.
Let X be a sparse × n matrix and I = I ( X ) . ( a ) The defining ideal of F ( I ) is generated by the Pl¨ucker relations on the × -minors of X and these form a Gr¨obner basis of the ideal they generate. ( b ) R ( I ) is of fiber type, that is, its defining ideal is generated by the relations of the sym-metric algebra of I and by the Pl¨ucker relations on the × -minors of X . Moreover,these equations form a Gr¨obner basis of the ideal they generate. ( c ) R ( I ) and F ( I ) have rational singularities in characteristic zero and they are F -rationalin positive characteristic. In particular, they are Cohen-Macaulay normal domains. ( d ) R ( I ) and F ( I ) are Koszul algebras. In particular, the powers of I have a linear resolu-tion. ( e ) The natural algebra generators of R ( I ) and F ( I ) are SAGBI bases of the algebras theygenerate. Notation
This section is devoted to the setup and notation we will use throughout the paper. Let K be a field, n ≥ X = ( x ij ) a 2 × n sparse matrix of indeterminatesover K , i.e., a matrix whose entries x ij are either distinct indeterminates or zero. We usethe notation x i,j when it is not clear what the two subscripts are. Let R = K [ X ] denote thepolynomial ring over K in the variables appearing in X and let I be the ideal generated bythe 2 × X . p to permuting the rows and the columns of X , we may assume that there exist r and s with 0 ≤ n − r − s ≤ r < n such that X = (cid:18) x , · · · x ,r x ,r +1 · · · x ,r + s · · · · · · x ,r +1 · · · x ,r + s x ,r + s +1 · · · x ,n (cid:19) . If r = 0, then n = s and X is a generic matrix. Since the results obtained in this paper areknown in this case, we may assume without loss of generality that r ≥
1. Let f ij denote the2 × X that involves columns i and j . Then I = ( f ij | ≤ i < j ≤ n ) ⊆ R. Notice that f ij = − f ji and that f ij is a monomial or zero, unless r + 1 ≤ i, j ≤ r + s .By the form of X , if a minor is non-zero, then the product of the entries on its diagonalis also non-zero. This means that we can choose a diagonal term order on K [ X ], that is, aterm order with the property that the leading term of each minor of X is the product of theelements on its diagonal. An example of a diagonal term order is the lexicographic orderwith x , > . . . > x ,r + s > x ,r +1 > . . . > x ,n . The maximal minors of X form a diagonalGr¨obner basis for I by [2, Proposition 5.4].Throughout the paper, we fix a diagonal term order τ . The minimal generating set forin τ ( I ) with respect to τ is described next. Proposition 2.1 ([2], Proposition 5.4) . Let X be a sparse × n matrix, I the ideal of × minors of X , and τ a diagonal term order. The initial ideal of I is in τ ( I ) = ( x i x j | ≤ i ≤ r + s, max { r, i } < j ≤ n ) . It turns out that in τ ( I ) corresponds to a Ferrers diagram as in Figure 1. x ,n · · · x ,r + s +1 x ,r + s · · · x ,r +1 x , ... x ,r x ,r +1 ... x ,r + s A BC D
Figure 1.
Ferrers diagram
Corollary 2.2.
The ideal in τ ( I ) is the edge ideal of a Ferrers bipartite graph whose vertexsets are V = { x , , . . . , x ,r + s } , W = { x ,n , . . . , x ,r +1 } , nd whose partition is λ = ( n − r, . . . , n − r | {z } r times , n − r − , n − r − , . . . , n − r − s ) . In other words, the first r elements of V have edges connecting them to all n − r elementsof W Moreover, for i > r , the i th element of V is connected by an edge to the first n − i elements of W . We say that in τ ( I ) is the Ferrers ideal I λ . See [13] for more on Ferrers graphs, diagrams,and ideals. Example 2.3.
For the matrix X = (cid:18) x x x x x x x x x x x x x (cid:19) the ideal in τ ( I ) is the Ferrers ideal I λ for the partition λ = (6 , , , , , , x x x x x x x x x x x x x Figure 2.
Ferrers diagram for λ = (6 , , , , , , Definition 2.4 ([13]) . The one-sided ladder associated to a Ferrers ideal I λ is the ladder L λ with the same shape as the Ferrers diagram for λ and for which the entry in row x i andcolumn x j is y ij .The ladder associated to in τ ( I ) = I λ is the two-sided ladder L ′ λ obtained from L λ byadding a row of n − r boxes at the top and a column of r + s boxes on the left. The newboxes are filled with the variables from W and V , respectively, in their listed order.Illustrations of L λ and L ′ λ for λ = (6 , , , , , ,
2) are in Figure 3. x x x x x x x x x x x x y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y x x x x x x x x x x x x x y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y Figure 3.
Ladders L λ and L ′ λ for λ = (6 , , , , , , x ij are only row and column markers in L λ , whereas in L ′ λ they are entriesof the ladder. The entries in L λ and in L ′ λ are distinct variables. Therefore, results for ladderdeterminantal ideals apply, see for instance [10, 33].3. Rees algebras of I and of its initial ideal The Rees algebra of an ideal can be realized as a quotient of a polynomial ring. When I is an ideal generated by n elements, say f , . . . , f n , we let y , . . . , y n be variables over R and consider the map from R [ y , . . . , y n ] to R ( I ) that maps y i to f i t . Hence R ( I ) ∼ = R [ y , . . . , y n ] /J , where J is the defining ideal of the algebra, and the defining equations are asystem of generators of J . The defining equations of the Rees algebra are in general difficultto compute or determine theoretically. The largest y -degree of a minimal generator of thedefining ideal J is the relation type of the ideal and plays an important role in the study ofblowup algebras. Finally, we say that the Rees algebra of I is of fiber type if the definingideal of the Rees algebra is generated by the defining equations of the symmetric algebraand the defining equations of the special fiber ring.Given a term order τ on a polynomial ring R over a field K , one can extend it to a termorder τ ′ on R [ t ] as follows. Let a, b ∈ R be monomials and let i, j be non-negative integers.Define(3.1) at i < τ ′ bt j ⇔ i < j or i = j and a < τ b. For a K -subalgebra A of R [ t ], one defines in τ ′ ( A ) as the K -algebra generated by all initialmonomials of elements in A . When A is homogeneous in the variable t , then in τ ′ ( A ) = ⊕ i ≥ in τ ′ ( A i ), where A i is the set of elements of A that are homogeneous of degree i in t . Inparticular, in our settingin τ ′ ( R ( I )) = ⊕ i ≥ in τ ′ ( R ( I ) i ) = ⊕ i ≥ in τ ( I i ) t i . ince R (in τ ( I )) = ⊕ i ≥ (in τ ( I )) i t i , in order to prove that in τ ′ ( R ( I )) = R (in τ ( I )), it sufficesto show that (in τ ( I )) i = in τ ( I i ) for all i ≥
1. A result by Conca, Herzog, and Valla statesthat it suffices to check the equality up to the relation type of in τ ( I ). Theorem 3.1. [12, Corollary 2.8]
Let R be a polynomial ring, I a homogeneous ideal in R ,and τ a term order on R . Suppose that (in τ ( I )) i = in τ ( I i ) for ≤ i ≤ reltype(in τ ( I )) . Then (in τ ( I )) i = in τ ( I i ) for all i ≥ and reltype( I ) ≤ reltype(in τ ( I )) . The equality in τ ′ ( R ( I )) = R (in τ ( I )) was established by Conca in [11, Theorem 2.1] forthe ideal I of maximal minors of a generic m × n matrix. The following lemma is the key toestablishing in τ ′ ( R ( I )) = R (in τ ( I )) in our case. Lemma 3.2.
Let X be a sparse × n matrix, I the ideal of × minors of X , and τ adiagonal term order on R = K [ X ] . Then in τ ( I ) = (in τ ( I )) .Proof. We proceed by induction on n ≥
2. For n = 2, the ideal I is principal and the resultholds automatically. To proceed we choose a decomposition of I as follows.Let I be the ideal generated by the variables x j for r + 1 ≤ j ≤ n and let I be the idealgenerated by the 2 × X obtained by deleting the first columnof X . Since r ≥
1, then I = x I + I . By Proposition 2.1, in τ ( I ) = x I + in τ ( I ). Noticehere that for I we may have that its corresponding r is 0. In that case in τ ( I ) = (in τ ( I )) by [11, Theorem 2.1]. Otherwise, the equality holds by induction.Certainly (in τ ( I )) ⊆ in τ ( I ). To prove equality it is enough to show that the Hilbertfunctions of (in τ ( I )) and I are the same, since the Hilbert function of I is the same asthe Hilbert function of in τ ( I ).By induction on n , the Hilbert functions of I and (in τ ( I )) are the same. We have I = ( x I + x I I ) + I and (in τ ( I )) = ( x I + x I in τ ( I )) + (in τ ( I )) . Both of these ideals are of the form J = ( x I + x I J ) + J , with J = I and J = I inthe former case and with J = in τ ( I ) and J = in τ ( I ) in the latter case. Each case gives riseto the short exact sequence:0 −→ ( x I + x I J ) ∩ J −→ ( x I + x I J ) ⊕ J −→ J −→ . (3.2)Since the variable x is a non-zerodivisor on J , and since J ⊆ I , the intersection on theleft-hand side of the sequence is:( x I + x I J ) ∩ J = ( x I + x I J ) ∩ ( x ) ∩ J = ( x I + x I J ) ∩ x J = x J . This means that the short exact sequence simplifies to0 −→ x J −→ ( x I + x I J ) ⊕ J −→ J −→ . (3.3)By the induction hypothesis, the two incarnations of J and hence of x J have the sameHilbert function, so that to prove that the two incarnations I and (in τ ( I )) of J have the ame Hilbert function it suffices to prove that the two incarnations of x I + x I J havethe same Hilbert function. As before we get a short exact sequence:0 −→ x I ∩ I J −→ x I ⊕ I J −→ x I + I J −→ , (3.4)and since x is a non-zerodivisor on I J and J ⊆ I , the short exact sequence simplifies to0 −→ x I J −→ x I ⊕ I J −→ x I + I J −→ . (3.5)The conclusion will follow once we show that in the two incarnations of J the Hilbertfunction of I J is the same, that is in τ ( I I ) = I in τ ( I ). Clearly, I in τ ( I ) ⊆ in τ ( I I ).Define a grading on R by setting deg( x i ) = 0 and deg( x j ) = 1 for all i, j . With this grading, I is generated by elements of degree 1, I is generated by elements of degree 1, and I I isgenerated by elements of degree 2. It suffices to show that for every homogeneous f ∈ I I ,in τ ( f ) ∈ I in τ ( I ). Clearly in τ ( f ) has degree at least 2 and is in in τ ( I ). By Proposition 2.1,in τ ( I ) is generated by elements of degree 1. Thus by degree reasons, to write in τ ( f ) as anelement of in τ ( I ), the coefficient must have degree at least 1, i.e., the coefficient must be in( x i | i = r + 1 , . . . , n ) = I . It follows that in τ ( f ) ∈ I in τ ( I ). (cid:3) We are now ready to prove the main theorem of this section. Let A be a subalgebra of apolynomial ring over a field K . We recall that a subset C of A is a SAGBI basis for A withrespect to a term order τ if in τ ( A ) is generated as a K -algebra by the initial monomials ofthe elements in C with respect to τ . In general, SAGBI bases are difficult to compute andmay not even be finite. When I is the ideal of maximal minors of a generic m × n matrix X , Conca showed that in τ ′ ( R ( I )) = R (in τ ( I )) with respect to a diagonal term order τ [11,Theorem 2.1]. Moreover, Narasimhan [33, Corollary 3.4] showed that the maximal minorsof X form a Gr¨obner basis for I . It then follows that the natural algebra generators of R ( I )are a SAGBI basis of it. The following theorem extends these results to the case of the idealmaximal minors of a sparse 2 × n matrix. Theorem 3.3.
Let X be a sparse × n matrix and let I be the ideal of × minors of X .Let τ be a diagonal term order on R = K [ X ] and let τ ′ be its extension to R [ t ] as in (3.1).Then in τ ′ ( R ( I )) = R (in τ ( I )) and reltype( I ) ≤ . Moreover, the set { x i , x j | ≤ i ≤ r + s, r + 1 ≤ j ≤ n } ∪ { f ij t | ≤ i < j ≤ n } is aSAGBI basis of R ( I ) with respect to τ ′ .Proof. According to [13, Proposition 5.1], the defining ideal of the special fiber ring of I λ =in τ ( I ) is the ideal generated by the 2 × L λ . By [41, Theorem 3.1] we know that R (in τ ( I )) is of fiber type and hence the relation type of in τ ( I ) is at most 2.By Lemma 3.2 we have that in τ ( I ) = (in τ ( I )) , and thus by Theorem 3.1, the relationtype of I is at most 2 and in τ ( I i ) = (in τ ( I )) i for all i . Therefore,in τ ′ ( R ( I )) = ⊕ i ≥ in τ ( I i ) = ⊕ i ≥ (in τ ( I )) i = R (in τ ( I )) . The last part of the claim now follows, since the 2 × X are a Gr¨obner basis of I by [2, Proposition 5.4]. Therefore the set { x i , x j | ≤ i ≤ r + s, r + 1 ≤ j ≤ n } ∪ { f ij t | ≤ i < j ≤ n } is a SAGBI basis of R ( I ), since the leading terms of these elements generate R (in τ ( I )) as K -algebra and in τ ′ ( R ( I )) = R (in τ ( I )). (cid:3) n immediate consequence of the equality in τ ( I i ) = (in τ ( I )) i is the following corollary. Corollary 3.4.
Let X be a sparse × n matrix and let I the ideal of × minors of X .Let τ be a diagonal term order on R = K [ X ] . The i -fold products of maximal minors of X are a τ -Gr¨obner basis of I i for every i . Using the theory of SAGBI bases, one can now deduce properties of the Rees algebra of I from those of its initial algebra. Corollary 3.5.
Let X be a sparse × n matrix and let I the ideal of × minors of X .Then R ( I ) has rational singularities if the field K has characteristic and it is F -rationalif K has positive characteristic. In particular, R ( I ) is a Cohen-Macaulay normal domain.Proof. Let τ be a diagonal term order on R = K [ X ] and let τ ′ be its extension to R [ t ] asin (3.1). By [36, Corollary 5.3 and Theorem 5.9], R (in τ ( I )) is a Cohen-Macaulay normal do-main. Thus in τ ′ ( R ( I )) is a Cohen-Macaulay normal domain by Theorem 3.3. The conclusionthen follows from [12, Corollary 2.3]. (cid:3) The Defining Equations of the Rees Algebra
In this section we obtain the defining equations of the Rees algebra of the ideal of 2 × × n matrix X . In addition to the setup and notation from Section 2,we will also adopt the following. Setting and Notation 4.1.
Let S = K [ L ′ λ ] and T = K [ L λ ] be the polynomial rings over K in the variables that appear in L ′ λ and L λ , respectively. To simplify our notation, we makethe identifications: • x ui = 0 if u = 1 and i > r + s or if u = 2 and i ≤ r , • y ij = 0 if i, j ∈ { , . . . , r } or i, j ∈ { r + s + 1 , . . . , n } ,to give meaning to all other x ui , y ij with u ∈ { , } , ≤ i < j ≤ n . Define the followingstandard presentations of the symmetric algebra S ( I ) , the Rees algebra R ( I ) of I , and thespecial fiber ring F ( I ) : σ : S −→ S ( I ) ,ρ : S −→ R ( I ) ,ϕ : T −→ F ( I ) , where for all i, j , σ ( x ij ) = ρ ( x ij ) = x ij , σ ( y ij ) = ϕ ( y ij ) = f ij , and ρ ( y ij ) = f ij t . We let L = ker( σ ) , J = ker( ρ ) , and K = ker( ϕ ) . The ideals L , J , K are called the defining idealsof the symmetric algebra, the Rees algebra, and the special fiber ring of I , respectively.We similarly define the presentations of the symmetric algebra, of the Rees algebra, andof the special fiber ring of in τ ( I ) : σ ′ : S −→ S (in τ ( I )) ,ρ ′ : S −→ R (in τ ( I )) = in τ ′ ( R ( I )) ,ϕ ′ : T −→ F (in τ ( I )) , here for all i, j , σ ( x ij ) = ρ ( x ij ) = x ij , σ ( y ij ) = ϕ ( y ij ) = in τ ( f ij ) , and ρ ( y ij ) = in τ ( f ij ) t .Moreover, let L ′ = ker( σ ′ ) , J ′ = ker( ρ ′ ) , and K ′ = ker( ϕ ′ ) denote the defining ideals of thesymmetric algebra, the Rees algebra, and the special fiber of in τ ( I ) , respectively. Remark 4.2.
Let ℓ uijk = x ui y jk − x uj y ik + x uk y ij and p ijkl = y ij y kl − y ik y jl + y il y jk for u ∈ { , } and 1 ≤ i < j < k < l ≤ n . With the identifications from the Setting andNotation 4.1, ℓ uijk , p ijkl are in S for all u ∈ { , } , 1 ≤ i < j < k < l ≤ n . We call the ℓ uijk ’s the linear relations of I and the p ijkl ’s the Pl¨ucker relations of I . Clearly ℓ uijk ∈ L and ℓ uijk , p ijkl ∈ J for all u ∈ { , } and 1 ≤ i < j < k < l ≤ n .In the next proposition we describe the defining equations of the symmetric algebra, thespecial fiber, and the Rees algebra of in τ ( I ). In addition, we show that the ℓ uijk ’s are indeedthe defining equations of the symmetric algebra of I . Proposition 4.3.
Adopt Setting and Notation 4.1. Then ( a ) L = ( ℓ uijk | u = 1 ,
2; 1 ≤ i < j < k ≤ n ) . ( b ) L ′ is generated by the × minors of L ′ λ that involve either the first column or the firstrow. ( c ) K ′ = I ( L λ ) . ( d ) J ′ = I ( L ′ λ ) = L ′ + I ( L λ ) S .Moreover, the natural generators of I ( L λ ) and I ( L ′ λ ) are Gr¨obner bases of K ′ and J ′ ,respectively, with respect to a diagonal term order.Proof. (a) By [2, Theorem 4.1 and its proof] the presentation matrix of I is obtained from theEagon-Northcott resolution by “pruning”. In particular, the relations on the generators f ij of I arise from taking the 3 × × n matrix obtained from X by doubling oneof the two rows. These yield precisely the relations on the symmetric algebra of the givenform.Item (b) follows from [26, Theorem 5.1] since Ferrers ideals satisfy the ℓ -exchange property,see e.g. [31, Lemma 6.3].(c) This follows from [13, Proposition 5.1].(d) Since in τ ( I ) is a Ferrers ideal, it is the edge ideal of a bipartite graph. Therefore theRees algebra of in τ ( I ) is of fiber type by [41, Theorem 3.1].The last part follows from the fact that I ( L λ ) and I ( L ′ λ ) are ladder determinantal ideals.Therefore, by [33, Corollary 3.4], the 2 × L λ and L ′ λ are Gr¨obner bases of I ( L λ )and I ( L ′ λ ), respectively, with respect to a diagonal term order. (cid:3) The description of the defining equations of the Rees algebra of in τ ( I ), in combinationwith our result from the previous section that shows that in τ ′ ( R ( I )) = R (in τ ( I )), allows usto deduce that the Rees algebra of I is a Koszul algebra. Corollary 4.4.
Let X be a sparse × n matrix, I the ideal of × minors of X . Then R ( I ) is a Koszul algebra and I has linear powers. In particular, reg( I k ) = 2 k for all k ∈ N . roof. The defining ideal of R (in τ ( I )) has a Gr¨obner basis of quadrics by Proposition 4.3,so R (in τ ( I )) is a Koszul algebra, see [21] and [20, Theorem 6.7]. By Theorem 3.3 we havein τ ( R ( I )) = R (in τ ( I )) and since R (in τ ( I )) is a Koszul algebra, then R ( I ) is a Koszulalgebra, by [12, Corollary 2.6]. According to Blum [3, Corollary 3.6], if the Rees algebra ofan ideal is Koszul, then the ideal has linear powers, i.e., the powers of the ideal have a linearresolution. (cid:3) We now are ready to prove the main result of this article.
Theorem 4.5.
Let X be a sparse × n matrix, I the ideal of × minors of X . AdoptSetting and Notation 4.1. ( a ) The defining ideal J of R ( I ) is generated by the linear relations ℓ uijk for u ∈ { , } and ≤ i < j < k ≤ n , and the Pl¨ucker relations p ijkl for ≤ i < j < k < l ≤ n . Moreover,these generators form a Gr¨obner basis of J with respect to a suitable weight. ( b ) The Rees algebra R ( I ) is of fiber type, i.e., J = L + K S. ( c ) The Pl¨ucker relations p ijkl for ≤ i < j < k < l ≤ n are the defining equations of F ( I ) .Proof. (a) We let G be the set of all ℓ uijk and p ijkl as in Remark 4.2. We claim that J = ( G ).Define a weight ω on R [ t ] and a weight π on S as follows: ω ( x j ) = 1 , ω ( x j ) = j, ω ( t ) = 1 ,π ( x j ) = 1 , π ( x j ) = j, π ( y ij ) = ω (in τ ( f ij ) t ) = j + 2 . By Proposition 4.3, I ( L ′ λ ) is the defining ideal of R (in τ ( I )) = in τ ′ ( R ( I )). The weight ω represents the term order τ , that is, in ω ( f ij ) = in τ ( f ij ). Therefore, by [38, Proposition inLecture 3.1] (see also [39, Theorem 11.4]) we conclude that I ( L ′ λ ) = in π ( J ).Since G ⊆ J , to prove that G is a Gr¨obner basis of J with respect to π , it suffices toprove that each 2 × L ′ λ is the leading form with respect to π of some element in G . This also implies that J is generated by G .We first analyze the 2 × L ′ λ that involve the first column. These are of theform E ijk = x i y jk − x j y ik with i < j < k and they are homogeneous of weight k + 3. Since x k y ij has weight j + 3, E ijk is the leading form of ℓ ijk ∈ G .We next analyze the 2 × L ′ λ that involve the first row. These are of the form E ijk = x k y ij − x j y ik with i < j < k and they are homogeneous of weight j + k + 2. Since x i y jk has weight i + k + 2, E ijk is the leading form of ℓ jik ∈ G .It remains to prove that each 2 × L λ is a unit multiple of a leading form of anelement in G . Such a minor is of the form F ijkl = y il y jk − y ik y jl for some i < j < k < l andit is homogeneous of weight k + l + 4. Since y ij y kl has weight j + l + 4, F ijkl is the leadingform of p ijkl ∈ G .Items (b) and (c) follow immediately from (a). (cid:3) The following corollary is a direct consequence of Theorem 4.5.
Corollary 4.6.
Let X be a sparse × n matrix, I the ideal of × minors of X , and τ a diagonal term order on R = K [ X ] . Then in τ ( F ( I )) = F (in τ ( I )) . Moreover, the × inors of X are a SAGBI basis of the K -algebra F ( I ) , the Pl¨ucker relations form a Gr¨obnerbasis for K , and F ( I ) is a Koszul algebra.Proof. By Proposition 4.3, the defining equations of F (in τ ( I )) are the 2 × L λ , i.e., F (in τ ( I )) = K [in τ ( f ij )] ∼ = T /I ( L λ ). For any i < j < k < l , let F ijkl = y il y jk − y ik y jl be agenerator of I ( L λ ). In the proof of Theorem 4.5 we showed that p ijkl = F ijkl + y ij y kl . Hence ϕ ( F ijkl ) = − ϕ ( y ij y kl ) and in τ ϕ ( F ijkl ) = in τ ϕ ( y ij y kl ). Then by [12, Proposition 1.1] thegenerators of I are a SAGBI basis of F ( I ), in particular in τ ( F ( I )) = F (in τ ( I )). Moreover,the Pl¨ucker relations form a Gr¨obner basis of K by [39, Corollary 11.6], hence F ( I ) is aKoszul algebra. (cid:3) Finally, we obtain the analogous result for F ( I ) as in Corollary 3.5. Corollary 4.7.
Let X be a sparse × n matrix and I the ideal of × minors of X . Then F ( I ) has rational singularities if the field K has characteristic and is F -rational if K haspositive characteristic. In particular, F ( I ) is a Cohen-Macaulay normal domain.Proof. By [36, Theorem 7.1] and Corollary 3.5, F (in τ ( I )) is normal. Thus, by [28, Theorem1] F (in τ ( I )) is Cohen-Macaulay as well. By Corollary 4.6, we have in τ ( F ( I )) = F (in τ ( I )).The result now follows from [12, Corollary 2.3]. (cid:3) Corollary 4.8.
Let X be a sparse × n matrix and I the ideal of × minors of X . Then ( a ) F ( I ) has dimension min { n + s − , n − r − } . ( b ) F ( I ) is Gorenstein if r ≤ . ( c ) reg( F ( I )) = min { n − , n − r − , r + s − } . ( d ) The a -invariant of F ( I ) is a ( F ( I )) = min {− r − s, r − n, − s − } , unless n = r + s inwhich case a ( F ( I )) = − n if r = 1 or a ( F ( I )) = − n + 1 otherwise.Proof. (a) The formula follows from [12, Corollary 2.6] in combination with [36] or [10,Section 4].(b) By [10, Proposition 2.5] F (in τ ( I )) is Gorenstein if and only if r ≤
2. The statementnow follows from [12, Corollary 2.3].(c) From [13, Proposition 5.7] we obtain the regularity of F (in τ ( I )), which simplifies toreg( F (in τ ( I ))) = min { n − , n − r − , r + s − } . By [12, Corollary 2.5] we have a ( F (in τ ( I ))) = a ( F ( I )) and since F (in τ ( I )) is Cohen-Macaulay, then reg( F ( I )) = reg( F (in τ ( I ))).(d) The result follows from the fact that a ( F ( I )) = − dim F ( I )+reg( F ( I )). Alternatively,the formula is a special case of [22, Corollary 9]. In the special case n = r + s , we havedim F ( I ) = 2 n − r − F ( I )) = min { n − , n − r − } . (cid:3) Corollary 4.9.
Let X be a sparse × n matrix and I the ideal of × minors of X . Let λ be the partition for the Ferrers ideal in τ ( I λ ) . Then the normalized Hilbert series of F ( I ) is p λ ( z ) = 1 + h ( λ ) z + . . . + h r + s − ( λ ) z r + s − , where h k ( λ ) = (cid:0) r + s − k (cid:1)(cid:0) n − r − k (cid:1) − (cid:0) s +1 k +1 (cid:1)(cid:0) n − k − (cid:1) with the convention that (cid:0) ji (cid:1) = 0 if j < i . Moreover, the multiplicity of F ( I ) is e ( F ( I )) = (cid:0) n + s − n − r − (cid:1) − (cid:0) n + s − n − (cid:1) . roof. First notice that the h -vectors for F ( I ) and F (in τ ( I )) coincide and in particular, e ( F ( I )) = e ( F (in τ ( I ))) by [12, Corollary 2.5]. Since F (in τ ( I )) is a ladder determinantalring the formula for the Hilbert series is obtained in [42, Theorem 4.7]. One can then deducethe formula for the multiplicity immediately. The formula for the multiplicity is also workedout explicitly in [43, Corollary 4.2]. (cid:3) Corollary 4.10.
Let X be a sparse × n matrix and I the ideal of × minors of X . Then ( a ) The regularity of R ( I ) is reg( R ( I )) = min { n − , n − r, r + s } . ( b ) The a -invariant of R ( I ) is a ( R ( I )) = min {− s − , − s − r − , r − n − } . ( c ) The normalized Hilbert series of R ( I ) is p λ ( z ) = 1 + h ( λ ) z + . . . + h r + s ( λ ) z r + s , where h ( λ ) = ( r + s )( n − r ) − (cid:0) s +12 (cid:1) − and h k ( λ ) = (cid:0) r + sk (cid:1)(cid:0) n − rk (cid:1) − (cid:0) s +1 k +1 (cid:1)(cid:0) n − k − (cid:1) for k = 1 , withthe convention that (cid:0) ji (cid:1) = 0 if j < i . ( d ) The multiplicity of R ( I ) is e ( R ( I )) = (cid:0) n + sn − r (cid:1) − (cid:0) n + sn +1 (cid:1) − .Proof. Let λ = ( n − r, . . . , n − r | {z } r times , n − r − , n − r − , . . . , n − r − s ) ,µ = ( n − r + 1 , . . . , n − r + 1 | {z } r +1 times , n − r, n − r − , . . . , n − r − s + 1) , and let I ( L ′ λ ) be the defining ideal of R (in τ ( I )). By [24, Theorem 2.1], the ideal I ( L µ ) isobtained from the ideal I ( L λ ) generated by the entries of L λ by an ascending G-biliaison ofheight 1 on I ( L ′ λ ). Specifically, one has(4.1) yI ( L µ ) + I ( L ′ λ ) = f I ( L λ ) + I ( L ′ λ ) , where y is the variable appearing in position (2 ,
2) of L µ and f is the 2 × L µ .(a) By [12, Corollary 2.5] we have a ( R (in τ ( I ))) = a ( R ( I )) and since R (in τ ( I )) and R ( I )are Cohen-Macaulay, then reg( R ( I )) = reg( R (in τ ( I ))). Since reg( K [ L µ ] /I ( L λ )) = 0, thenreg( R ( I )) = reg( K [ L µ ] /I ( L µ )) = min { n − , n − r, r + s } , where the first equality follows from [17, Theorem 3.1] and the second from Corollary 4.8.(b) The formula for the a -invariant of R ( I ) now follows from the fact that a ( R ( I )) = − dim R ( I ) + reg( R ( I )) = min {− s − , − s − r − , r − n − } .We now prove (c) and (d) together. The normalized Hilbert series of R ( I ) and R (in τ ( I ))coincide and in particular e ( R ( I )) = e ( R (in τ ( I ))) by [12, Corollary 2.5]. Using the shortexact sequences0 −→ I ( L ′ λ )( − −→ I ( L µ )( − ⊕ I ( L ′ λ ) −→ yI ( L µ ) + I ( L ′ λ ) −→ −→ I ( L ′ λ )( − −→ I ( L λ )( − ⊕ I ( L ′ λ ) −→ f I ( L λ ) + I ( L ′ λ ) −→ HS K [ L µ ] /I ( L µ ) ( z ) = zHS K [ L µ ] /I ( L λ ) ( z ) + (1 − z ) HS K [ L µ ] /I ( L ′ λ ) ( z ) , here HS A ( z ) denotes the Hilbert series of the algebra A . Therefore, by Corollary 4.9 h ( λ ) = h ( µ ) − r + s )( n − r ) − (cid:18) s + 12 (cid:19) − h k ( λ ) = h k ( µ ) = (cid:18) r + sk (cid:19)(cid:18) n − rk (cid:19) − (cid:18) s + 1 k + 1 (cid:19)(cid:18) n − k − (cid:19) for k = 1 , where 1 + h ( µ ) z + . . . + h r + s ( µ ) z r + s is the normalized Hilbert series of K [ L µ ] /I ( L µ ). Inparticular, the multiplicity of R (in τ ( I )) is e ( R (in τ ( I ))) = e ( K [ L µ ] /I ( L µ )) − e ( K [ L µ ] /J ) = (cid:18) n + sn − r (cid:19) − (cid:18) n + sn + 1 (cid:19) − . We close with the following remark that gives an alternative path for a proof of our resultsfor special types of m × n sparse matrices. Remark 4.11.
Let X be an m × n sparse matrix, m ≤ n . We assume that, after row andcolumn permutations, the variables that appear in X form a two-sided ladder as in Figure 4. Figure 4.
A matrix with a shaded two-sided ladder.The case of a sparse 2 × n matrix X is a special case of the above type of matrix with m = 2, where the ladder has one lower inside corner in position (1 , r + 1) and one upperinside corner in position (2 , r + s ). Let I = I m ( X ) and let τ be a diagonal term order.Notice that an m × m -minor of X is non-zero if and only if all the entries on its diagonal arenon-zero. Moreover, the m × m -minors of X form a universal Gr¨obner basis of I m ( X ) by [2,Proposition 5.4]. It follows that in τ ( I ) is generated by the products of the elements on thediagonals of the m × m non-zero minors of X .One can show that in τ ( I ) is a sortable ideal with respect to the lexicographic order inducedby the following order of the variables: x i,j > x k,l if either i < k or i = k and j < l . Therefore,the defining ideal of F (in τ ( I )) is the toric ideal generated by the binomial relations obtainedby the sorting [20, Theorem 6.16]. Moreover, these generators are a quadratic Gr¨obner basisof the defining ideal of F (in τ ( I )). In particular, the sorting we use allows us to conclude hat in τ ( I ) is a generalized Hibi ideal and F (in τ ( I )) is a generalized Hibi ring, see [27] and[20, Section 6.3]. Therefore, F (in τ ( I )) is a normal, Cohen-Macaulay, Koszul algebra.Furthermore, in τ ( I ) is a weakly polymatroidal ideal with respect to the same order onthe variables as above. For the definition of a weakly polymatroidal ideal, see [29] or [20,Definition 6.25]. Hence, by [20, Proposition 6.26] in τ ( I ) satisfies the ℓ -exchange propertywith respect to the sorting order. Therefore, R (in τ ( I )) is of fiber type and the definingequations of R (in τ ( I )) are precisely the defining equations of the special fiber ring F (in τ ( I ))and the linear relations by [20, Theorem 6.24].Because of the special shape of our matrix, one can use the criterion in [35] (see also [12,Proposition 1.1]) and proceed in a similar manner as in [20, Theorem 6.46] to show thatthe Pl¨ucker relations are the defining equations of F ( I ) and the maximal minors of X area SAGBI basis of F ( I ). One can also show that Pl¨ucker relations along with the linearrelations of S ( I ) are the defining equations of R ( I ) and the maximal minors of X along withthe variables of R are a SAGBI basis of R ( I ). In particular in τ ′ ( R ( I )) = R (in τ ( I )) andin τ ( F ( I )) = F (in τ ( I )), where τ ′ is a suitable term order. One then obtains similar resultsfor R ( I ) and F ( I ) as in Corollaries 3.5, 4.4, 4.6, and 4.7. Acknowledgements.
We are grateful to the Women in Commutative Algebra (WICA)group for organizing the first “Women in Commutative Algebra” workshop at BIRS, Banff,Canada, where this project began. We also thank the staff at BIRS for their great hospitalityduring the workshop. The WICA workshop was funded by the National Science Foundationgrant DMS 1934391 and by the Association for Women in Mathematics grant NSF-HRD1500481.
References [1] A. Almousa, K.-N. Lin, and W. Liske, Rees algebras of closed determinantal facet ideals, available at arXiv:2008.10950. [math.AC] .[2] A. Boocher, Free resolutions and sparse determinantal ideals, Math. Res. Lett. (2012), 805–821.[3] S. Blum, Subalgebras of bigraded Koszul algebras, J. Algebra (2001), no.2, 795–809.[4] W. Bruns, A. Conca, and M. Varbaro, Maximal minors and linear powers, J. Reine Angew. Math. (2015), 41–53.[5] W. Bruns, A. Simis, and N. Trung, Blow-up of straightening-closed ideals in ordinal hodge algebras,Trans. Amer. Math. Soc. (1991), 507–528.[6] W. Bruns and U. Vetter, Determinantal rings, Lecture Notes in Math. , Springer-Verlag BerlinHeidelberg (1988).[7] L. Bus´e, On the equations of the moving curve ideal of a rational algebraic plane curve, J. Algebra (2009) 2317–2344.[8] L. Bus´e and J.-P. Jouanolou, On the closed image of a rational map and the implicitization problem,J. Algebra (2003), no. 1, 312–357.[9] F. Chen, W. Wang, and Y. Liu, Computing singular points of plane rational curves, J. Symbolic Comput.43 (2008), 92–117.[10] A. Conca, Ladder Determinantal Rings, J. Pure Appl. Algebra (1995), 119–134.[11] A. Conca, Gr¨obner bases of powers of ideals of maximal minors, J. Pure Appl. Algebra (1997),223–231.[12] A. Conca, J. Herzog, and G. Valla, Sagbi bases with applications to blow-up algebras, J. Reine Angew.Math. (1996), 113–138.[13] A. Corso and U. Nagel, Monomial and toric ideals associated to Ferrers graphs, Trans. Amer. Math.Soc. (2009), 1371–1395.
14] D. Cox, The moving curve ideal and the Rees algebra, Theoret. Comput. Sci. (2008), no. 1-3, 23–36.[15] D. Cox, K.-N. Lin, and G. Sosa, Multi-Rees algebras and toric dynamical systems, Proc. Amer. Math.Soc. (2019), 4605–4616.[16] C. De Concini, D. Eisenbud, and C. Procesi, Hodge algebras, Ast´erisque, (1982).[17] E. De Negri and E. Gorla, Invariants of ideals generated by Pfaffians, Commutative Algebra and ItsConnections to Geometry, A. Corso and C. Polini Eds., Contemporary Mathematics (2011), 47–62.[18] D. Eisenbud, Introduction to algebras with straightening laws, Ring Theory and Algebra, III - Proc.Third Conf., Univ. Oklahoma, Norman, Okla., 1979, Lecture Notes in Pure and Appl. Math. , Dekker,New York (1980), 243–268.[19] D. Eisenbud and C. Huneke, Cohen-Macaulay Rees algebras and their specialization, J. Algebra (1983), 202–224.[20] V. Ene and J. Herzog, Gr¨obner bases in commutative algebra, Graduate Studies in Mathematics ,American Mathematical Society, Providence, RI (2012).[21] R. Fr¨oberg, Koszul algebras, Advances in commutative ring theory (Fez, 1997), 337–350, Lecture Notesin Pure and Appl. Math. , Dekker, New York (1999).[22] S. R. Ghorpade and C. Krattenthaler, Computation of the a -invariant of ladder determinantal rings, J.Algebra Appl. (2015), no. 9, 1502001-1–1502001-26.[23] M. Giusti and M. Merle, Singularit´es isol´ees et sections planes de vari´et´es d´eterminantielles II - Sectionsde vari´et´es d´eterminantielles par les plans de coordonn´ees, Algebraic Geometry (La R´abida, 1981),103–118 (1982).[24] E. Gorla, Mixed ladder determinantal varieties from two-sided ladders, J. Pure Appl. Algebra ,(2007), 433–444.[25] E. Gorla, J. Migliore, and U. Nagel, Gr¨obner bases via linkage, J. Algebra (2013), 110–134.[26] J. Herzog, T. Hibi, and M. Vladoiu, Ideals of fiber type and polymatroids, Osaka J. Math., (2005),807–829.[27] J. Herzog and T. Hibi, Distributive lattices, bipartite graphs and Alexander duality, J. Algebraic Com-bin. (2005), 289–302.[28] M. Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes,Ann. of Math. , 318–337.[29] M. Kokubo and T. Hibi, Weakly polymatroidal ideals, Alg. Colloq. (2006), 711–720.[30] D. Kapur, K. Madiener, A completion procedure for computing a canonical basis for a k -subalgebra, inComputer and Mathematics (Cambridge MA 1989), Springer (1989), 1–11.[31] K.-N. Lin and Y.-H. Shen, Koszul blow-up algebras associated to three-dimensional Ferrers diagrams,J. Algebra, (2018), 219–253.[32] K.-N. Lin and Y.-H. Shen, Blow-up algebras of rational normal scrolls and their secant varieties, availableat arXiv:2009.03484. [math.AC] .[33] H. Narasimhan, The irreducibility of ladder determinantal varieties, J. Algebra (1986) 162–185.[34] D. Rees, Two classical theorems of ideal theory, Proc. Cambridge Philos. Soc. (1956), 155–157.[35] L. Robbiano and M. Sweedler, Subalgebra bases, Commutative Algebra - Proceedings of a workshopheld in Salvador, Brazil, Aug. 8–17, 1988 Eds. W. Bruns, A. Simis, Lect. Notes Math. , 61–87,Springer (1990).[36] A. Simis, W. Vasconcelos, and R.H. Villarreal, On the ideal theory of graphs, J. Algebra (1994),389–416.[37] N. Song, F. Chen, and R. Goldman, Axial moving lines and singularities of rational planar curves,Comput. Aided Geom. Design 24 (2007), 200–209.[38] B. Sturmfels, Toric ideals, Preliminary Lecture Notes, Workshop on Commutative Algebraand its relation to Combinatorics and Computer Algebra, ICPT Trieste (1994), available at http://indico.ictp.it/event/a02275/contribution/1 .[39] B. Sturmfels, Gr¨obner bases and convex polytopes, University Lecture Series , American MathematicalSociety (1996).[40] N. V. Trung, On the symbolic powers of determinantal ideals, J. Algebra (1979), 361–369.[41] R. Villarreal, Rees algebras of edge ideals, Comm. in Alg. (1995), 3513–3524.
42] H.-J. Wang, A determinantal formula for the Hilbert series of determinantal rings of one-sided ladder,J. Algebra (2003), no. 1, 79–99.[43] H.-J. Wang, Counting of paths and the multiplicity of determinantal rings, available at arXiv:0210375 [math.AC] . Ela Celikbas, Department of Mathematics, West Virginia University, Morgantown, W.V.26506
Email address : [email protected] URL : https://math.wvu.edu/~ec0029 Emilie Dufresne, Department of Mathematics, University of York, York, UK
Email address : [email protected] URL : Louiza Fouli, Department of Mathematical Sciences, New Mexico State University, LasCruces, NM 88003
Email address : [email protected] URL : Elisa Gorla, Institut de Math´ematiques, Universit´e de Neuchˆatel, Rue Emile-Argand 11,CH-2000 Neuchˆatel, Switzerland
Email address : [email protected] URL : http://members.unine.ch/elisa.gorla/ Kuei-Nuan Lin, Department of Mathematics, Penn State University, Greater Alleghenycampus, McKeesport, PA 15132
Email address : [email protected] URL : https://sites.psu.edu/kul20 Claudia Polini, Department of Mathematics, University of Notre Dame, 255 Hurley,Notre Dame, IN 46556
Email address : [email protected] URL : Irena Swanson, Department of Mathematics, Purdue University, 150 N. University Street,West Lafayette, IN 47907
Email address : [email protected] URL :