Blow-up algebras of rational normal scrolls and their secant varieties
BBLOW-UP ALGEBRAS OF RATIONAL NORMAL SCROLLS ANDTHEIR SECANT VARIETIES
KUEI-NUAN LIN AND YI-HUANG SHEN
Abstract.
We show that the fiber cones of general rational normal scrolls are Cohen–Macaulay and compute their Castelnuovo–Mumford regularities. Then we study thesecant varieties of balanced rational normal scrolls. We describe the defining equationsof their associated Rees algebras and compute the Castelnuovo–Mumford regularities oftheir fiber cones. Introduction
In this paper, we will study the blow-up algebras of rational normal scrolls and theirsecant varieties. Recall that if R = K [ x , . . . , x n ] is a polynomial ring over a field K , and I is an ideal of R minimally generated by elements of the same degree, then the Reesalgebra R ( I ) := ⊕ i ≥ I i t t ⊆ R [ t ], and the fiber cone F ( I ) = R ( I ) ⊗ R K ∼ = K [ I ] ⊆ R ,where t is a new variable. The Rees algebra and the fiber cone give the homogeneouscoordinate rings of the graph and image of the blow-up of the projective space along thesubscheme defined by I .We are mostly interested in the case when the ideal I is generated by the maximalminors of an (extended) Hankel matrix and describes the corresponding rational nor-mal scroll or its secant variety. As determinantal varieties, rational normal scrolls andtheir secant varieties are central objects in the study of algebraic varieties. The study ofdeterminantal varieties has attracted earnest attentions of algebraic geometers and com-mutative algebraists, partly due to the beautiful structures involved and the interestingapplications to the applied mathematics and statistics; see, for instance, [6], [9], [15] and[39], to name but a few. Rees algebra and its special fiber play a prominent role in thecommutative algebra field, as they encode the asymptotic behavior of the ideals definingthem; see, for example, [42]. One of the most challenging questions is to describe thosealgebras in term of generators and relations; see, for instance, [21], [27], [28], [30] and[31]. This is actually a classical and difficult question in the elimination theory. It isalso known in the applied mathematics fields such as geometry modelings (in the formof the moving curve theory, [12] and [13]) and chemical reaction networks ([14]). Oncewe know the relations among the generators, we can study the classical properties andinvariants of those algebras such as the Cohen–Macaulayness, the normality, the rationalsingularities, and the Castelnuovo–Mumford regularity; see, for instance, the work [1],[2], [7], and [29], which are related to the blow-up algebras of determinantal varieties.In the following, let n , . . . , n d be a sequence of positive integers and c = (cid:80) di =1 n i .For each i = 1 , . . . , d , let C i ⊆ P c + d − be a rational normal curve of degree n i withcomplementary linear spans and let ϕ i : P → C i be the corresponding isomorphism. Mathematics Subject Classification . Primary 13C40, 13A30, 13P10, 13F50; Secondary 14N07,14M12 .Keyword: Rational normal scroll, Secant variety, Rees algebra, Fiber cone, Regularity, Cohen–Macaulay. a r X i v : . [ m a t h . A C ] S e p hence, the related rational normal scroll is simply S n ,...,n d := (cid:91) p ∈ P { ϕ ( p ) , ϕ ( p ) , . . . , ϕ d ( p ) } ⊆ P c + d − . It is well-known ([24]) that S n ,...,n d is uniquely determined by the sequence n , . . . , n d upto projective equivalence, and in suitable coordinates the ideal of S n ,...,n d is generated bythe maximal minors of the matrix X := Å x , x , · · · x ,n − x , x , · · · x ,n − · · · x d, x d, · · · x d,n d − x , x , · · · x ,n x , x , · · · x ,n · · · x d, x d, · · · x d,n d ã . Now, let I = I ( X ) be the ideal generated by maximal minors of X . Sammartano gavea complete description of the defining equations of the blow-up algebras in [38], writingexplicitly the blow-up algebras R ( I ) and F ( I ) as the quotient rings of some polynomialrings. Meanwhile, when | n i − n j | ≤ i, j , Conca, Herzog and Valla showed in [8]that R ( I ) and F ( I ) are Cohen–Macaulay normal domains. Furthermore, if char( K ) = 0,then R ( I ) and F ( I ) have rational singularities. And if char( K ) >
0, then R ( I ) and F ( I )are F -rational. The assumption | n i − n j | ≤ balanced in the literature.In this work, we will apply the result of Sammartano to show the Cohen–Macaulaynessof F ( I ) in Theorem 2.1, without assuming the balanced condition. In particular, thisrecovers the corresponding work of Conca, Herzog and Valla in [8, Theorem 3.8 andProposition 4.6] for the balanced case. After that, we also compute the Castelnuovo–Mumford regularity of F ( I ) in Theorem 5.2. The Cohen–Macaulayness just establishedpaves the way, since we can resort to the balanced case by [3, Theorem 3.7] and [4,Proposition 7.43].In addition, we can extend the number of rows of the matrix X and study the blow-upsof the maximal minors. When the new matrix is balanced, we can rewrite it as the r × c extended Hankel matrix H r,c,d := à x x x · · · x c x d x d x d · · · x c + d ... ... ... ... x r − d x r − d x r − d · · · x c +( r − d í over the ring R = K [ x , . . . , x c +( r − d ]. This kind of matrix is also called d -leap catalec-ticant or d -catalecticant in [37]. Nam showed in [33, Corollary 3.9] that I r ( H r,c,d ), theideal of the maximal minors of H r,c,d , defines the ( r − I ( H ,c +( r − d,d ). It is then natural to ask whether we can ex-tend the results of Conca, Herzog and Valla in [8], finding the defining equations of theblow-up algebras, and establishing the expected properties of those algebras. As a mat-ter of fact, we are able to use the Sagbi basis theory to obtain the Gr¨obner bases of thedefining ideals of those algebras, and obtain the normality and the Gorensteinness of thefiber cone F ( I r ( H r,c,d )). Finally, we can compute explicitly the Castelnuovo–Mumfordregularity of the fiber cone.The outline of the paper is the following. In section 2, we show in Theorem 2.1 thatthe fiber cone defined by the rational normal scroll is Cohen–Macaulay. To achieve that,we associate the initial ideal of the defining ideal of the fiber cone obtained by the work[38] of Sammartano, with its Stanley–Reisner complex. We will show that the ideal of theAlexander dual of this complex has linear quotients. Equivalently, the original simplicialcomplex is shellable and consequently Cohen–Macaulay. Hence, the fiber cone is Cohen–Macaulay. To establish the linear-quotients property, we have to resort to the binarytree structure of the facets of the simplicial complex, established by Sammartano. To be ore accurate, instead of just using the lexicographic order on the minimal monomialgenerators of the Alexander dual ideal, we need to group these generators first using theleaves of the binary tree; see Notation 2.4. The subsequent proof is also quite involved,since we have to analyze the binary tree structure in detail, and pin down those lineargenerators of the successive colon ideals.In section 3, we turn our focus to the blow-ups of the ( r − R ( I r ( H r,c,d )) and the fibercone F ( I r ( H r,c,d )). Gr¨obner bases of those two algebras are obtained in Theorem 3.9and in Corollary 3.8, respectively, using the Sagbi basis theory. In particular, we showthat the maximal minors of the matrix H r,c,d induce a Sagbi basis of the fiber cone inTheorem 3.7. As a by-product, we derive the normality and rational singularity of thefiber cone in Proposition 3.10. Moreover, we can characterize the Gorensteinness of thefiber cone. As a reminder, Nam showed in [33] that the initial algebra of the Rees algebrais actually the Rees algebra of the initial ideal. But he didn’t consider explicitly theinitial algebra of the fiber cone. And the defining equations of the Rees algebra or thefiber cone are not described explicitly in [33]. Our biggest contribution in this part is thenatural algorithm in Proposition 3.5 to construct the defining ideals of those two blow-upalgebras of the I r ( H r,c,d ).In section 4, we use the explicit description of the initial ideal to compute the Castelnuovo–Mumford regularity of the fiber cone in Theorem 4.1. The formula presented there israther dazzling, let alone its more involved proof. To be brief, we first associate thegenerators of the Alexander dual of the initial ideal of the defining ideal, with the max-imal cliques of some graph G in the subsection 4.1. The main combinatorial tool is themoving sequence introduced in Observation 4.8. This notion is essential for investigatingthe generators of the Alexander dual ideal comprehensively. Then, we consider simulta-neously the lexicographic and reverse lexicographic types of ordering with respect to theground ring R = K [ x , . . . , x c +( r − d ]. We will show that both orderings give complete-intersection-quotients structures of the Alexander dual ideal in the subsection 4.2. Moreprecisely, each of the successive colon ideals is either a collection of single variables (cor-ner generators) or a collection of single variables with one extra higher degree monomial(tail generator). Even with complete intersection quotients, we only obtain a good upperbound on the regularity so far. In order to write down the precise value, we use themapping cones to extract the projective dimension of the Alexander dual ideal, whichin turn gives the regularity of the fiber cone. However, due to the existence of the tailgenerator, we don’t have linear quotients in general, i.e., the mapping cones won’t give aminimal free resolution of the Alexander dual ideal. Thus, we need to apply lots of care-ful observations, intricate techniques and relatively prudent strategies during the process.The biggest obstacle lies in the relatively degenerated case when the number of columns c of the matrix is smaller relative to the number of rows r and the leaping distance d .Whence, the maximal length of the complete intersection quotients is much harder tojustify; see subsection 4.5. To overcome it, we will create a balance by employing bothlexicographic and reverse lexicographic types of ordering simultaneously. As an applica-tion, we obtain the reduction number of I r ( H r,c,d ) and the a -invariant of F ( I r ( H r,c,d ))in Corollary 4.22.Finally, in section 5, we compute the Castelnuovo–Mumford regularity of the fibercone of general rational normal scrolls in Theorem 5.2. Its argument depends on theCohen–Macaulayness established in section 2 and the regularity result of balanced casesin section 4. . Cohen–Macaulayness of the fiber cones of rational normal scrolls
Throughout this paper, as usual, if n is a positive integer, [ n ] will be the set { , , . . . , n } .Recall that a rational normal scroll is uniquely determined by a sequence of positive in-tegers n = ( n , ..., n d ) with n ≤ · · · ≤ n d by [24]. Let c := n + n + · · · + n d . In suitablecoordinates, the defining ideal I = I n ,...,n d of the corresponding rational normal scroll S n ,...,n d ⊆ P c + d − is generated by the maximal minors of the matrix X where X := ( X , X , . . . , X d )= ( x , x , · · · x ,n − x , x , · · · x ,n − · · · x d, x d, · · · x d,n d − x , x , · · · x ,n x , x , · · · x ,n · · · x d, x d, · · · x d,n d ) . Here is the main result of this section.
Theorem 2.1.
Let I n ,...,n d be the defining ideal of the rational normal scroll S n ,...,n d ⊆ P c + d − for c = n + · · · + n d . Then the fiber cone F ( I n ,...,n d ) is Cohen–Macaulay. We need some preparations before presenting its proof. Let us start by recalling theindispensable constructions in [38] for the fiber cone F ( I ) where I = I n ,...,n d . First of all,a matrix M = M n ,...,n d was introduced in [38, Section 2] as follows. One starts with thefirst column of the i -th catalecticant block X i of the original matrix X = ( X , X , . . . , X d )for each i increasingly in i , then the second column, and so on until one has used allcolumns except the last one for each block; when a block runs out of columns one simplyskips it. The first c − d columns of M form the submatrix Ç x i (2) , x i (2)+1 , · · · x d, x i (3) , x i (3)+1 , · · · x d, x i (4) , · · · · · · x d,n d − x i (2) , x i (2)+1 , · · · x d, x i (3) , x i (3)+1 , · · · x d, x i (4) , · · · · · · x d,n d − å , where the i ( l ) denotes the least integer i such that n i ≥ l . The last d columns of M areof the last column of the i -th block of X for each i , but ordered decreasingly in i : Ç x d,n d − x d − ,n d − − · · · x ,n − x ,n − x d,n d x d − ,n d − · · · x ,n x ,n å . Interested readers are invited to go through the examples in [38, Example 2.1].Let π : K [ T α,β | ≤ α < β ≤ c ] (cid:16) F ( I )be the ring epimorphism determined by T α,β (cid:55)→ det( M α,β ), where M α,β is the 2 × M using the α -th and β -th columns. Then P := ker( π ) is called the definingideal of the fiber cone F ( I ). Under suitable monomial ordering introduced in [38], theinitial ideal in( P ) is squarefree and quadratic. One can then consider its associatedStanley–Reisner complex ∆, which will be called the initial complex of the fiber cone F ( I ) following [38]. Proof of Theorem 2.1.
When c < d + 4, the fiber cone is the coordinate ring of theGrassmann variety G (1 , c − ⊂ P ( c ) − ; see [38, Remark 3.15]. Thus, the fiber cone isCohen–Macaulay by [26].Now, it remains to consider the case when c ≥ d + 4. Notice that F ( I ) being Cohen–Macaulay is equivalent to saying that the defining ideal P has this property. By [25,Corollary 3.3.5], or more strongly, by [10, Corollary 2.7], it suffices to show that in( P ) isCohen–Macaulay. But this in turn is equivalent to showing that in( P ) ∨ , the Alexanderdual, has a linear resolution by [25, Theorem 8.1.9]. Because of this, we will show inProposition 2.6 that in( P ) ∨ has linear quotients. This is sufficient for our purpose by[25, Proposition 8.2.1]. (cid:3) n the following, we shall assume that c ≥ d +4. Before we really start proving the linearquotients property, we still need some preparations. Under the natural identification T α,β ↔ ( α, β ), the vertex set of the aforementioned simplicial complex ∆ is V = { ( α, β ) | ≤ α < β ≤ c } . Following [38], the vertices of ∆ will also be described as open intervals in the real line R with integral endpoints. And we will use the familiar notions of length, intersection andcontainment of intervals.With respect to the above matrix M , for each α ∈ [ c − d −
2] and each i ∈ [ d ], let γ α,i be the least index γ ≥ α + 2 such that the γ -th column of M involves variables from theblock X i . Then, there exists some (cid:96) α with 2 ≤ (cid:96) α ≤ d + 1 such that { γ α, , . . . , γ α,d } = { α + 2 , . . . , α + (cid:96) α } ∪ { c − d + (cid:96) α , . . . , c } ; (1)c.f. [38, Definition 3.6]. The key observation that we shall apply is the following result. Lemma 2.2 ([38, Proposition 3.8]) . Suppose that c ≥ d + 4 . A subset F ⊆ V is afacet of ∆ if and only if the Hasse diagram T F of the poset ( F, ⊆ ) satisfies the followingconditions: (i) T F is a rooted binary tree with the root (1 , c ) ; (ii) there exists some α ∈ [ c − d − such that the leaves of T F are the intervals { ( β, β + 1) | β ∈ { α, . . . , α + (cid:96) α } ∪ { c − d + (cid:96) α − , . . . , c − } } ; (2)(iii) if I ∈ F is a node of T F with only one child I , then length( I ) = length( I ) + 1 and the unique unitary interval in I \ I does not belong to F ; (iv) if I ∈ F is a node of T F with two children I , I , then I ∩ I = ∅ and length( I )+length( I ) = length( I ) .In this case we have | F | = c + d . In the following, when we say the node ( α (cid:48) , β (cid:48) ) is a left child of the node ( α (cid:48)(cid:48) , β (cid:48)(cid:48) ) in thebinary tree T F , we mean it is a child and α (cid:48) = α (cid:48)(cid:48) . The notions of right child , left sibling and right sibling can be similarly defined.We will call the set in (2) as the leaves set of F and denote it by L α . Notice that it isuniquely determined by its smallest point α ∈ [ c − d −
2] for the given matrix M . Thefollowing easy fact will be needed later. Lemma 2.3.
For each α ∈ [ c − d − , the cardinalities | L α \ L α +1 | = | L α +1 \ L α | = 1 .Proof. Recall that for each such α and each i ∈ [ d ], the integer γ α,i is the least index γ ≥ α + 2 such that the γ -th column of M involves variables from the block X i . Furthermore,the union in (1) is actually disjoint. Therefore, γ α +1 ,i ≥ γ α,i . And when γ α,i ≥ ( α + 1) + 2,then γ α +1 ,i = γ α,i . Now, suppose that α + 2 = γ α,i . Then for i (cid:54) = i , we have γ α,i = γ α +1 ,i .Whence, L α +1 \ L α = { ( γ α +1 ,i , γ α +1 ,i + 1) } and L α \ L α +1 = { ( α, α + 1) } . (cid:3) It is time to introduce the expected linear ordering. Let S = K [ T v | v ∈ V ] be the basering. Endow it with the lexicographic order > lex with respect to the linear order on thevariables T v ’s such that T v > T v ⇔ the leftmost nonzero component of v − v is negative;here, by abuse of notation, we also think of V as a set of integral points on Z . Recallthat the facet F of the initial complex ∆ is bijectively related to the monomial (cid:100) T F := (cid:89) v ∈ V \ F T v n the minimal monomial generating set G ((in( P )) ∨ ) by [25, Lemma 1.5.3]. Therefore,to impose a linear order on G ((in( P )) ∨ ) amounts to giving a linear order of the facets.For later reference, facets with the common leaves set given by (2) will be grouped intothe family G α . Notation 2.4.
The expected linear order on the facets, denoted by (cid:31) , will be as follows.(a) If two facets F ∈ G α and F (cid:48) ∈ G α (cid:48) respectively with α > α (cid:48) , then F (cid:31) F (cid:48) .(b) Next, consider the facets F and F (cid:48) within the same group G α . Then F (cid:31) F (cid:48) ifand only if (cid:100) T F > lex ‘ T F (cid:48) . Example 2.5.
The first facet of the group G α with respect to our ordering (cid:31) is the onewith vertices (1 , c ) , (2 , c ) , . . . , ( c − , c )in addition to the leaves given in (2).For example, let n = (2 , , , M = Ç x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , å . If we choose α = 2, then the leaves set is { (2 , , (3 , , (4 , , (5 , , (10 , , (11 , } ;see also [38, Example 3.9]. The collection of open intervals in the Figure 1 is the firstfacet with respect to the given α = 2 in the corresponding complex ∆. And the Hasse1 2 3 4 5 6 7 8 9 10 11 12 Figure 1.
The first facet in G α when α = 2 and n = (2 , , , P )) ∨ has linear quotients. Proposition 2.6.
When c ≥ d + 4 , the ideal (in( P )) ∨ has linear quotients with respectto the given ordering (cid:31) .Proof. Let F be a facet of ∆ and suppose that F ∈ G α . We want to show that theminimal monomial generating set G F := G ( I F ) of the colon ideal I F := (cid:104) ‘ T F (cid:48) | F (cid:48) (cid:31) F (cid:105) : (cid:100) T F is linear. Notice that (cid:104) ‘ T F (cid:48) (cid:105) : (cid:100) T F = T F \ F (cid:48) := (cid:89) v ∈ F \ F (cid:48) T v . Suppose that F is the first facet within G α with respect to the designated linear order (cid:31) . If α = c − d −
2, then F is actually the first facet of ∆ and there is nothing to show Figure 2.
Binary tree of the first facet in G α when α = 2 and n = (2 , , , α < c − d −
2. Let F (cid:48) be the first facet of G α +1 . It is clear that F (cid:48) (cid:31) F and F \ F (cid:48) = L α \ L α +1 = { ( α, α + 1) } by Lemma 2.3 and Example 2.5. Notice that for any facet F (cid:48)(cid:48) preceding F , ( α, α + 1) ∈ F \ F (cid:48)(cid:48) . Whence, the colon ideal I F is principal and linear.Suppose instead that F is not the first facet in G α . This situation is more involved.We will break the long proof into three parts, according to the following plans.(1) Firstly, we describe the expected linear generating set G F .(2) Next, we show that the variables described above really show up as some colonsof monomials, and they are the only variables possessing this property.(3) Finally, we show that we don’t have minimal monomial generators in G F of higherdegrees.After showing the above three parts, it is clear then that the ideal (in( P )) ∨ has linearquotients. At this point, one may suggest only proving the last step. But to successfullygive the proof, we have to clear higher-degree generators by the linear ones.2.1. Linear generators of the colon ideal I F . We separate the vertices in F intocollections for each fixed facet F ∈ G α . For b ∈ [ c − F b := { v ∈ F | v = ( b, ∗ ) } . (3)If the cardinality k b := | F b | ≥
1, we may assume that F b = { ( b, i b, ) , ( b, i b, ) , . . . , ( b, i b,k b ) } with i b, < i b, < · · · < i b,k b . It is clear that ( b, i b,j +1 ) is the parent node of ( b, i b,j ) when1 ≤ j < k b . Now, assume that k b ≥
1, and we describe whether T b,i b,j ∈ G F for j ∈ [ k b ].(a) If j = k b , then T b,i b,kb / ∈ G F .(b) Suppose that 2 ≤ j < k b . If(i) the node at ( b, i b,j ) has a right sibling in T F , or(ii) the node at ( b, i b,j ) has two children in T F ,then T b,i b,j ∈ G F . Otherwise, T b,i b,j / ∈ G F .(c) Suppose that j = 1 < k b .(i) Suppose that ( b, i b, ) is a leaf in T F , namely that i b, = b + 1.(1) If b (cid:54) = α , then T b,i b, / ∈ G F .(2) Suppose that b = α , i.e., ( b, i b, ) is the leftmost leaf in T F . If b = c − d −
2, then G α is the first group to be considered. Whence,we expect T b,i b, / ∈ G F . • Otherwise, we expect T b,i b, = T α,α +1 ∈ G F and α (cid:54) = c − d + 2.(ii) Otherwise, i b, > b + 1. Whence, we always have T b,i b, ∈ G F .For later reference, we denote the set of the expected linear generators of F with respectto b described above as LG F,b . Then in the next two subsections, we will see that theminimal generating set G F is precisely the set LG F := (cid:91) b ∈ [ c − LG F,b . Example 2.7.
Consider the case when n = ( n , n , n ) = (2 , , F = { (1 , , (1 , , (1 , , (1 , , (1 , , (1 , , (1 , , (1 , , (2 , , (3 , , (4 , , (4 , , (9 , , (10 , } is a facet of the associated complex ∆. Consider b = 1. The subset F b consists of thevertices written on the first line. And our criterion above says that the correspondinglinear generators for b = 1 are LG F, = { T , , T , , T , , T , , T , } ⊆ LG F . Indeed, it is not difficult to see that equality holds in this special case.2.2.
Why the vertex of F shows or not shows up in G F ? Before we proceed, noticefirst the following consequences of Lemma 2.2.
Observations 2.8.
Let F be a facet of the aforementioned initial complex ∆. Recallthat it is convenient to consider the vertex ( α, β ) in ∆ as an open interval in the real line R with integral end points.(i) Vertices of F are non-crossing , i.e., if we consider the vertices I , I of F as openintervals, then I ∩ I = ∅ , or I ⊆ I , or I ⊆ I .(ii) If ( α, β ) ∈ F is not the root and has no sibling, then its parent node is either( α − , β ) or ( α, β + 1). Correspondingly, ( α − , α ) or ( β, β + 1) is not a leaf of T F .(iii) If as an open interval, ( α, β ) contains no leaf of T F , then the vertex ( α, β ) doesnot belong to F .(iv) If both ( α, β ) and ( α, β (cid:48) ) belong to F with β < β (cid:48) such that ( β, β (cid:48) ) contains noleaf of T F , then ( α, β + j ) ∈ F for all j with 0 < j < β (cid:48) − β . To see this, noticethat for each such fixed j , F must have at least one vertex of the form ( β + j, γ ) or( α (cid:48) , β + j ). In the first subcase, γ ≤ β (cid:48) is impossible by the previous observation.But when γ > β (cid:48) , this is also impossible, since the two open intervals ( β + j, γ )and ( α, β (cid:48) ) cross. As for the second subcase, by the previous observation, it isclear that α (cid:48) < β . If α < α (cid:48) < β , then the two open intervals ( α (cid:48) , β + j ) and ( α, β )cross. And if α (cid:48) < α , then the two open intervals ( α (cid:48) , β + j ) and ( α, β (cid:48) ) cross.Therefore, α (cid:48) = α and ( α, β + j ) is a vertex of F , as expected.(v) Suppose that ( γ, γ + 1) ∈ V is not a leaf of T F . Consider the smallest interval(node) in T F that contains ( γ, γ + 1). Then by Lemma 2.2 and the minimality,this node is either ( γ (cid:48) , γ + 1) or ( γ, γ (cid:48) ) for some γ (cid:48) . And correspondingly it has aunique child node ( γ (cid:48) , γ ) or ( γ + 1 , γ (cid:48) ). nother fact is that the variable T v ∈ G F if and only if F \ F (cid:48) = { v } for some F (cid:48) (cid:31) F .And for this pair F and F (cid:48) , if F \ F (cid:48) = { ( i, j ) } while F (cid:48) \ F = { ( i (cid:48) , j (cid:48) ) } , theneither i (cid:48) > i or i (cid:48) = i and j (cid:48) > j .Now, we are ready to justify the linear generating set LG F presented in the previoussubsection 2.1. Correspondingly, we also have three cases.(a) If j = k b , we show that T b,i b,kb / ∈ G F . We don’t have to worry about the case when b = 1, since (1 , c ) is the common root. Suppose now for contradiction that thereis some F (cid:48) with F (cid:48) (cid:31) F and F \ F (cid:48) = { ( b, i b,k b ) } . Thus, we have two subcases.Suppose first that F ∈ G α and F (cid:48) ∈ G α (cid:48) with α < α (cid:48) . Since necessarily ( α, α +1) ∈ L α \ L α (cid:48) ⊆ F \ F (cid:48) while | F \ F (cid:48) | = 1, it follows that b = α and i b,k b = α + 1.Since the leftmost leaf of T F is ( α, α + 1) and | F α | = 1, F must contain the vertex( α − , α + 1) by the item (ii) of Observations 2.8. Hence, F (cid:48) must also containthis vertex. But the leftmost leaf of T F (cid:48) is ( α (cid:48) , α (cid:48) + 1) with α (cid:48) > α , contradictingthe item (iii) of Observations 2.8.Suppose now instead that F and F (cid:48) have the same leaves set, i.e., α = α (cid:48) . Then( b, i b,k b ) is not a leaf of T F and must have some child. The parent node of ( b, i b,k b )in T F must have the form ( b (cid:48) , i b,k b ) for some b (cid:48) < b . Notice that the unique vertexin F (cid:48) \ F that the vertex ( b, i b,k b ) is switched to cannot take the form ( b, b (cid:48)(cid:48) ) with b (cid:48)(cid:48) > i b,k b , since the interval ( b, b (cid:48)(cid:48) ) will cross the interval ( b (cid:48) , i b,k b ). Meanwhile, thisunique vertex cannot take the form ( b (cid:48)(cid:48) , b (cid:48)(cid:48)(cid:48) ) with b (cid:48)(cid:48) > b , since ( b (cid:48) , i b,k b ) will nothave a legal branch in T F (cid:48) ; see (iii) and (iv) of Lemma 2.2. Therefore, we have acontradiction to the fact stated at the end of Observations 2.8.Thus, T b,i b,kb / ∈ G F .(b) Suppose that 2 ≤ j < k b .(i) Suppose that the node ( b, i b,j ) has a sibling in T F . It is clear that the parentnode of ( b, i b,j ) is ( b, i b,j +1 ) and ( b, i b,j ) is the left child in the binary tree.Whence, the sibling of ( b, i b,j ) is ( i b,j , i b,j +1 ). Now, let F (cid:48) := F ∪ { ( i b,j − , i b,j +1 ) } \ { ( b, i b,j ) } . Obviously F (cid:48) (cid:31) F . And it is easy to verify that F (cid:48) is a legal facet andthe node ( b, i b,j +1 ) has two children ( b, i b,j − ) and ( i b,j − , i b,j +1 ) in T F (cid:48) . Thisimplies that T b,i b,j ∈ G F .(ii) Suppose that the node ( b, i b,j ) has two children in T F . They have to be( b, i b,j − ) and ( i b,j − , i b,j ). We similarly define F (cid:48) := F ∪ { ( i b,j − , i b,j +1 ) } \ { ( b, i b,j ) } , and see that T b,i b,j ∈ G F .(iii) Suppose that the node ( b, i b,j ) has only one child and has no right sibling.Whence, i b,j ± = i b,j ± i b,j − , i b,j +1 ) contains no leaf in T F by Observations 2.8 (ii). Suppose for contradiction that we have some facet F (cid:48) such that F (cid:48) (cid:31) F and F \ F (cid:48) = { ( b, i b,j ) } . Since ( b, i b,j ) is not a leaf, F and F (cid:48) have identical leaves set. Therefore ( i b,j − , i b,j +1 ) contains no leaf in T F (cid:48) as well. The node ( b, i b,j − ) must be a grandchildren of ( b, i b,j +1 ) in T F (cid:48) by Observations 2.8 (iv) and Lemma 2.2 (iii) and (iv). Meanwhile, ( b, i b,j )is the only possible option to connect ( b, i b,j − ) with ( b, i b,j +1 ). This makes F = F (cid:48) , a contradiction. Therefore, T b,i b,j / ∈ G F .(c) Suppose that j = 1 < k b . i) We first consider the case when ( b, i b, ) is a leaf in T F , namely that i b, = b +1.Suppose that T b,i b, ∈ G T . Then we have F ∈ G α and F (cid:48) ∈ G α (cid:48) for some α < α (cid:48) such that F \ F (cid:48) = { ( b, b + 1) } . It is clear then that α < α (cid:48) ≤ c − d −
2. Andagain, since ( α, α + 1) ∈ L α \ L α (cid:48) ⊆ F \ F (cid:48) while | F \ F (cid:48) | = 1, it follows that b = α .Conversely, suppose that b = α < c − d − F ∈ G α . Notice that L α +1 \ L α = { ( β, β + 1) } for some β by Lemma 2.3, and this ( β, β + 1) / ∈ F .Meanwhile, since the invariant (cid:96) α defined before (1) satisfies (cid:96) α ≥
2, both( α, α + 1) and ( α + 1 , α + 2) are leaves of T F . Thus, as k α ≥
2, the node( α, α + 1) has a right sibling ( α + 1 , i α, ) by Lemma 2.2 (iii). It is clear thatany vertex in T F strictly containing ( α, α + 1) will contain its parent node( α, i α, ), and consequently will contain ( α + 1 , i α, ). Now, it is not difficult toverify that F (cid:48) := F ∪ { ( β, β + 1) } \ { ( α, α + 1) } ∈ G α +1 is a well-defined facet, and therefore, T α,α +1 ∈ G F .(ii) Next, we assume that ( b, i b, ) is not a leaf in T F , namely that i b, > b +1. Since j = 1, ( b, b + 1) is not a leaf and ( b, i b, ) has only the right child ( b + 1 , i b, )in T F . We have two subcases. • Suppose that ( b, i b, ) has a right sibling in T F . It has to be ( i b, , i b, ). • Suppose that ( b, i b, ) does not have a right sibling in T F . Whence, i b, = i b, + 1 and ( i b, , i b, + 1) is not a leaf of T F by Observations 2.8(ii).In either subcase, let F (cid:48) := F ∪ { ( b + 1 , i b, ) } \ { ( b, i b, ) } , and we can see as above that T b,i b, ∈ G F . And this completes our argumentin this subsection.2.3. No minimal monomial generator of higher degrees.
In this subsection, wefinish the proof by showing that G F = LG F .Suppose for contradiction that F ∈ G α and F (cid:48) ∈ G α (cid:48) with F (cid:48) (cid:31) F such that (cid:104) ‘ T F (cid:48) (cid:105) : (cid:100) T F provides a minimal monomial generator of higher degree in G F \ LG F . Whence, none ofthe linear generators in LG F will ever divide the principal generator of (cid:104) ‘ T F (cid:48) (cid:105) : (cid:100) T F . If weapply the identification T i,j ↔ ( i, j ) ∈ ∆, then this is just LG F ∩ ( F \ F (cid:48) ) = ∅ , or more simply, LG F ⊆ F ∩ F (cid:48) . (4)If α (cid:54) = α (cid:48) , let a ≤ α be the smallest such that ( a, α + 1) ∈ F . We claim that( a, α + 1) ∈ LG F . It is clear when a = α since α < α (cid:48) ≤ c − d −
2. Thus, we will assume a < α . The parent node of ( a, α + 1) has the form either ( a (cid:48) , α + 1) for some a (cid:48) < a , or( a, α (cid:48)(cid:48) ) for some α (cid:48)(cid:48) > α + 1. The first case contradicts the the minimality of a . Thus,we have the second case and consequently | F a | ≥
2. Whence, ( a, α + 1) ∈ LG F , stillestablishing the claim. Meanwhile, since α (cid:48) > α , ( a, α + 1) does not contain any leaf of T F (cid:48) . This implies that ( a, α + 1) ∈ F \ F (cid:48) , contradicting the assumption in (4).Therefore, we will assume in the following that α = α (cid:48) , i.e., T F and T F (cid:48) have a commonleaves set. Now, let b ∈ [ c −
1] be the smallest such that F b (cid:54) = F (cid:48) b . Here, F (cid:48) b is a subset of F (cid:48) , just as F b defined for F in (3).(a) Suppose that k b = 1. By the minimality of b and the fact that ‘ T F (cid:48) > lex (cid:100) T F , wehave ( b, j ) / ∈ F (cid:48) b for all j with b + 1 ≤ j ≤ i b, . Therefore, either ( b, i (cid:48) ) ∈ F (cid:48) b forsome i (cid:48) > i b, or F (cid:48) b = ∅ . n the first subcase, notice that the parent node of ( b, i b, ) in T F must take theform ( b (cid:48) , i b, ) for some b (cid:48) < b . By the minimality of b , ( b (cid:48) , i b, ) ∈ F (cid:48) . But since( b, i (cid:48) ) will cross ( b (cid:48) , i b, ) when considered as open intervals, this is impossible.In the remaining subcase, one can conclude that F (cid:48) contains some vertices of theform ( b (cid:48) , b +1) and ( b (cid:48) , b ) such that ( b, b +1) is not a leaf of T F (cid:48) , by Observations 2.8(v). Since b (cid:48) < b , we also have ( b (cid:48) , b + 1) , ( b (cid:48) , b ) ∈ F by the minimality of b .Meanwhile, as T F and T F (cid:48) have a common leaves set, ( b, b + 1) is not a leaf of T F as well. Whence, i b, > b + 1 and we have two crossing open intervals ( b (cid:48) , b + 1)and ( b, i b, ) for F , which is another contradiction.(b) When k b = 2, we observe first that ( b, i b, ) ∈ F ∩ F (cid:48) . To see this, notice thatif ( b, i b, ) ∈ F \ F (cid:48) , then ( b, i b, ) is not a common leaf. Whence, ( b, i b, ) ∈ LG F by our description in subsection 2.1. Now, we have ( b, i b, ) ∈ LG F ∩ ( F \ F (cid:48) ),contradicting the assumption in (4).Therefore, by the minimality of b and the fact that ‘ T F (cid:48) > lex (cid:100) T F , we have twosubcases: either ( b, i (cid:48) ) ∈ F (cid:48) b for some i (cid:48) > i b, or F (cid:48) b = { ( b, i b, ) } . One can likewiseexclude the first subcase using the non-crossing argument. Whence, we only needto check with the F (cid:48) b = { ( b, i b, ) } case. In T F (cid:48) , the parent node of ( b, i b, ) musttake the form of ( b (cid:48) , i b, ) for some b (cid:48) < b . By the minimally of b , ( b (cid:48) , i b, ) belongsto F ∩ F (cid:48) and will consequently be the parent node of ( b, i b, ) in both T F and T F (cid:48) .But the parent node of ( b, i b, ) in T F is ( b, i b, ), still a contradiction.(c) Assume now that k b ≥
3. If ( b, i b,j ) ∈ F (cid:48) b for all 1 ≤ j ≤ k b −
1, then we can argueas in the previous case (b). Thus, we have some j with 1 ≤ j ≤ k b − b, i b,j ) / ∈ F (cid:48) b . Let j be the smallest with this property. Notice that • we always have ( b, i b, ) ∈ F (cid:48) as in (b), and • for j with 2 ≤ j ≤ k b −
1, if ( b, i b,j ) has a sibling or two children, then( b, i b,j ) ∈ LG F ⊂ F (cid:48) by (4).Thus, 2 ≤ j ≤ k b − b, i b,j ) has neither a right child nor a right sibling in T F .As a matter of fact, ( b, i b,j ) has no right sibling in T F for j ≤ j ≤ k b −
1. To seethis, suppose for contradiction that there exists some j > j such that ( b, i b,j ) hasa right sibling. Let j be the smallest. Therefore, for all j with j ≤ j < j , ( b, i b,j )has no right sibling. Notice that ( b, i b,j − ) also has no right sibling. Therefore,( i b,j − , i b,j ) contains no leaf of T F and i b,j = i b,j − + j − ( j −
1) for j ≤ j ≤ j by Observations 2.8 (ii). Since F, F (cid:48) ∈ G α , it follows that ( i b,j − , i b,j ) contains noleaf of T F (cid:48) . Meanwhile, as ( b, i b,j ) has a right sibling in T F , ( b, i b,j ) ∈ LG F ⊂ F (cid:48) as well. Since we have assumed that ( b, i b,j − ) ∈ F (cid:48) , this forces ( b, i b,j ) ∈ F (cid:48) for j − < j ≤ j by Observations 2.8 (iv), contradicting the choice of j .Now, ( b, i b,j ) has no right sibling in T F for j − ≤ j ≤ k b −
1. It follows fromObservations 2.8 (ii) that the interval ( i b,j − , i b,k b ) has no leaf of T F , and i b,j = i b,j − + j − ( j −
1) for j − ≤ j ≤ k b . (5)Furthermore, as ( b, i b,j − ) ∈ F (cid:48) while ( b, i b,j ) / ∈ F (cid:48) , we have( b, i b,j ) ∈ F \ F (cid:48) for j ≤ j ≤ k b (6)by Observations 2.8 (iv).So far, ( b, i b, ) ∈ F ∩ F (cid:48) and ( b, i b,j ) ∈ F (cid:48) b for all j = 2 , , . . . , j − j . Therefore, we have { ( b, i b,j ) | ≤ j ≤ j − } ⊆ F (cid:48) b . (7) f the containment in (7) is strict, then we have some ( b, i (cid:48) ) ∈ F (cid:48) b \ F b . Since F (cid:48) (cid:31) F , i (cid:48) > i b,j . And by (5) with (6), we must have i (cid:48) > i b,k b . If b = 1,then ( b, i b,k b ) is the common root (1 , c ). The existence of such an i (cid:48) is impossible.Therefore, b > b, i b,k b ) in T F must have the form ( b (cid:48) , i b,k b )for some b (cid:48) < b . By the minimality of b , ( b (cid:48) , i b,k b ) ∈ F (cid:48) . Whence, we will arrive ata contradiction due to crossing. Therefore, the containment in (7) is actually anequality: { ( b, i b,j ) | ≤ j ≤ j − } = F (cid:48) b . Now, we consider the parent node of ( b, i k,j − ) in T F (cid:48) and get a similar contra-diction due to crossing. And this completes the proof in this subsection.In summary, the ideal (in( P )) ∨ has linear quotients, as expected. (cid:3) Corollary 2.9.
When c ≥ d + 4 , the initial complex ∆ is shellable.Proof. It follows from [25, Proposition 8.2.5] and Proposition 2.6. (cid:3) Defining equation of the balanced case
Throughout this section and the next one, d will be a fixed positive integer. And themain object will be the extended Hankel matrix of r rows and c columns H r,c,d := à x x x · · · x c x d x d x d · · · x c + d ... ... ... ... x r − d x r − d x r − d · · · x c +( r − d í over the ring R = K [ x ] = K [ x , . . . , x c +( r − d ] with 2 ≤ r ≤ c , considered as well in [33].This matrix is called balanced since after rearranging the columns, one can obtain theblock matrix x x d x d ··· x x d x d ··· ··· x d x d + d x d +2 d ··· x d x d x d ··· x d x d x d ··· ··· x d + d x d +2 d x d +3 d ··· ... ... ... ... ... ... ... ... ... x r − d x rd x r +1) d ··· x r − d x rd x r +1) d ··· ··· x d +( r − d x d + rd x d +( r +1) d ··· . If we write c = ad + c with c < d , then in this decomposition, there are d blocks, wherethe first c blocks have a + 1 columns each and the last c = c − c blocks have a columnseach. In the following, we will always write N = N ( r, c, d ) := dim( R ) = c + ( r − d and I = I r ( H r,c,d ). When r = 2, I = I ( H r,c,d ) gives the defining ideal of the balancednational normal scroll. More generally, when r > I r ( H r,c,d ) defines the ( r − I ( H ,c +( r − d,d ) by [33, Corollary 3.9].This section and the next one together can be considered as a natural continuationand application (with necessarily some extra effort) of the beautiful paper by Nam [33].Since that paper provides many interesting tools and results that we shall apply here, wewill provide a succinct review in the following.Denote by > lex the lexicographic monomial order on R induced by the order of thevariables x > x > · · · > x N . We will only use this term order on R .It is clear that every maximal minor of H r,c,d can be uniquely determined by the indicesof the elements on its main diagonal: α < α < · · · < α r . Thus, we will denote thisminor by M ( α ) = M ( α , α , . . . , α r ). For instance, the minor using the first r columnswill be M (1 , d, d, . . . , r + ( r − d ). Notice that in this increasing sequence, one as α i + d < α j for i = 1 , , . . . , r −
1. This leads to the following partial order < d on theset of positive integers: i < d j if and only if i + d < j. We say that a sequence of positive integers α , α , . . . , α s is a < d -chain if α < d α < d · · · < d α s . Similarly, we say that a monomial x α · · · x α s is a < d -chain if its indices, whenorganized increasingly, form a < d -chain. Note that a monomial of degree r in R is a < d -chain if and only if it is the initial monomial of a maximal minor of H r,c,d .In the following, we will writeΛ r,d ( N ) := { α = ( α , . . . , α r ) | ≤ α < d α < d · · · < d α r ≤ N } . For each α = ( α , . . . , α r ) ∈ Λ r,d ( N ), we also write x α := x α · · · x α r . It is clear that I = (cid:104) M ( α ) | α ∈ Λ r,d ( N ) (cid:105) . Proposition 3.1 ([33, Corollary 3.9]) . With respect to the lexicographic order on R , themaximal minors of H r,c,d provide a Gr¨obner basis of the ideal I . In particular, the initialideal in > lex ( I ) is generated by G r,c,d := { x α | α ∈ Λ r,d ( N ) } . (8)In addition to Proposition 3.1, one actually has the following much stronger result. Proposition 3.2 ([33, Theorem 3.28(b)]) . One has (in > lex ( I )) k = in > lex ( I k ) for all posi-tive integer k . Temporarily, we fix a term order τ on the monomials in R . Recall that if A is afinitely generated K -subalgebra of R , the initial algebra of A , denoted by in τ ( A ), is the K -subalgebra of R generated by the initial monomials in τ ( a ) for all a ∈ A . And a setof elements a i ∈ A , i ∈ I , is called a Sagbi basis if in τ ( A ) = K [in τ ( a i ) | i ∈ I ]. Theterminology “Sagbi” is the acronym for “Subalgebra analog to Gr¨obner bases for ideal”.In this work, the term order τ in mind is the lexicographic order > lex .We need to extend the term order > lex on R to the order (cid:31) on R [ t ] = K [ x , t ] asfollows: for two monomials x a t i and x b t j of R [ t ], set x a t i (cid:31) x b t j if i > j or i = j and x a > lex x b with respect to the lexicographic order. Here, x a := x a · · · x a N N for a = ( a , . . . , a N ) ∈ N N , and x b is similarly defined. Using Proposition 3.2 and [8,Theorem 2.7], one obtains the equality R (in > lex ( I )) = in (cid:31) ( R ( I )). One can then verifythat { x , . . . , x N } ∪ { M ( α ) t | α ∈ Λ r,d ( N ) } forms a Sagbi basis of the Rees algebra R ( I ) = R [ It ] = K [ x , . . . , x N , It ] ⊂ R [ t ].The most pleasant thing here is that we can use R (in > lex ( I )) to study the Rees algebra R ( I ) via the techniques in [8, Section 2]. For R (in > lex ( I )), we consider the followingcanonical epimorphism R [ Y ] := R [ Y α | α ∈ Λ r,d ( N )] → R (in > lex ( I )) , Y α (cid:55)→ in > lex ( M ( α )) t = x α t. The kernel of this homomorphism will be called the defining ideal of the Rees algebra R (in > lex ( I )). Proposition 3.3 ([33, Proposition 5.12 and Theorem 5.13]) . The Rees algebra R ( I ) = (cid:76) i ≥ I i t i ⊂ R [ t ] is a normal Cohen–Macaulay Koszul domain, defined by a Gr¨obner basisof quadratics. Furthermore, the initial algebra in (cid:31) ( R ( I )) = R (in > lex ( I )) is defined by aGr¨obner basis of quadratics such that the underlined part will give the leading monomial: (i) Y α Y β − Y α (cid:48) Y β (cid:48) : ( α (cid:48) , β (cid:48) ) is the quasi-sorted pair reduction of ( α , β ) ; (ii) x t Y α − x a k Y β : with α k − < d t < α k for some k and β := ( α , . . . , α k − , t, α k +1 , . . . , α r ) by assuming that a = −∞ . n particular, the quadratic equations in item (i) form a Gr¨obner basis of the definingideal of the fiber cone of in > lex ( I ) by [33, Theorem 5.14]. Actually, the reduction processthat Nam applied in [33, Section 5] aims at the more complicated multi-Rees algebra.However, when focusing on the Rees algebra here, it can be greatly simplified as followsto give a reduced Gr¨obner basis of quadratics.First of all, from the algorithm stated in [33, Algorithm 5.9], it is clear that the quasi-sorted pair ( α (cid:48) , β (cid:48) ) in the item (i) is actually sorted in the sense of [33, Definition 5.1].The notation of sortedness has actually been studied earlier. Let Mon r be the set ofmonomials of degree r in R = K [ x , . . . , x N ]. We will apply the sorting operator sort : Mon r × Mon r → Mon r × Mon r , ( u, v ) (cid:55)→ ( u (cid:48) , v (cid:48) )considered in [41, Chapter 14]. Recall that if u and v are two monomials in Mon r such that uv = x i x i · · · x i r with i ≤ i ≤ · · · ≤ i r , then u (cid:48) = x i x i · · · x i r − while v (cid:48) = x i x i · · · x r . And the pair ( u, v ) is called sorted if sort( u, v ) = ( u, v ).The existence of the generators in item (i) actually implies the following fact. Lemma 3.4.
The set G r,c,d defined in (8) is sortable, i.e., sort( G r,c,d × G r,c,d ) ⊆ G r,c,d × G r,c,d . Conversely, by [19, Theorem 6.16], the set G r,c,d being sortable also implies that thegenerators in item (i) is the reduced Gr¨obner basis of the defining ideal of K [in > lex ( I )] ∼ = F (in > lex ( I )), and the underlined part is the corresponding leading monomial. Lemma 3.4has been verified directly in [20, Proposition 3.1].Now, we are ready to state the following straightening-law type of result. Proposition 3.5.
Let M ( α ) and M ( β ) be two arbitrary maximal minors in I = I r ( H r,c,d ) .Then there exist α , . . . , α n , β , . . . , β n ∈ Λ r,d ( N ) and µ , . . . , µ n ∈ K , such that M ( α ) M ( β ) = n (cid:88) i =1 µ i M ( α i ) M ( β i ) , (9) with in > lex ( M ( α ) M ( β )) = in > lex ( M ( α ) M ( β )) > in > lex ( M ( α ) M ( β )) > · · · > in > lex ( M ( α n ) M ( β n )) , (10) and the pair ( x α i , x β i ) is sorted for each i . (11) Proof.
Let f = M ( α ) M ( β ) ∈ I . Suppose that sort( α , β ) = ( α , β ). By Lemma 3.4, α , β ∈ Λ r,d ( N ). Now, we write f := f − M ( α ) M ( β ) ∈ I and choose µ = 1. If f (cid:54) = 0, by Proposition 3.2, we can find α (cid:48) , β (cid:48) ∈ Λ r,d ( N ) and µ ∈ K such that in( f ) = µ x α (cid:48) x β (cid:48) . Now, as above, we can find α , β ∈ Λ r,d ( N ) with sort( α (cid:48) , β (cid:48) ) = ( α , β ) andmake f := f − µ M ( α ) M ( β ) ∈ I . If f (cid:54) = 0, we will keep the construction process.This process, however, will terminate due to the monomial ordering. And the propertiesstated in the proposition are clear. (cid:3) Remark 3.6.
The straightening law with only (9) and (10) above is hardly surprising.The real contribution here is the tail part in (10) as well as (11). Notice that with thelexicographic order on R , the initial monomial of each f i is easy to obtain. Since thesorting operator only cares about the product of the inputting pair, we can apply itto in > lex ( f i ) and find the next product of maximal minors with ease. And Lemma 3.4guarantees that this operation is well-defined. e also want to point out that Nam in [33] did not provide the explicit equations ofthe defining ideals of the Rees algebra R ( I ) and the fiber cone F ( I ). Not only that, hedid not study explicitly the Sagbi basis of the fiber cone. Theorem 3.7.
The set { M ( α ) | α ∈ Λ r,d ( N ) } forms a Sagbi basis of F ( I ) , i.e., onehas in > lex ( F ( I )) = F (in > lex ( I )) .Proof. We know that ¶ Y α Y β − Y α Y β | ( x α , x β ) = sort( x α , x β ) © gives a generating setof the defining ideal of the fiber cone F (in > lex ( I )) ∼ = K [in > lex ( M ( α )) | α ∈ Λ r,d ( N )] by[33, Theorem 5.14]. Equation (9) in Proposition 3.5 gives M ( α ) M ( β ) − M ( α ) M ( β ) = (cid:80) ni =2 µ i M ( α i ) M ( β i ). Now with (10), we can apply [8, Proposition 1.1]. (cid:3) Corollary 3.8.
The defining ideal P ⊂ K [ Y γ | γ ∈ G r,c,d ] of the fiber cone K [ I ] has areduced Gr¨obner basis given by Y α Y β − n (cid:88) i =1 µ i Y α i Y β i (12) for all unsorted pair ( α , β ) in G r,c,d × G r,c,d with x α > lex x β , such that M ( α ) M ( β ) = n (cid:88) i =1 µ i M ( α i ) M ( β i ) as in (9) . Moreover, { Y α Y β | ( x α > lex x β ) is unsorted } is the generating set of the initialideal of P with respect to some term order τ .Proof. It follows from Theorem 3.7 and [8, Corollary 2.1] that the polynomials in (12) willgenerate the defining ideal P . As for the remaining part, it suffices to apply Lemma 3.4,[19, Theorem 6.16] and [8, Corollary 2.2]. (cid:3) Similar to Proposition 3.5 and Corollary 3.8, we can apply [8, Corollary 2.2] to givea primitive lifting of the binomial for the syzygy part in item (ii) of Proposition 3.3.However, there is a more elegant way to do it, i.e., we can borrow the strength from theEagon–Northcott resolution in [18].
Theorem 3.9.
The relations in (12) together with the syzygy-type relations r +1 (cid:88) j =1 ( − j +1 x c j +( k − d Y c ,c + d,...,c j − +( j − d,c j +1 +( j − d,...,c r +1 +( r − d for arbitrary c < c < · · · < c r < c r +1 and arbitrary k with ≤ k ≤ r , form a Gr¨obnerbasis of the defining ideal of R ( I ) with respect to some term order on R [ Y ] .Proof. The presentation matrix of I can be obtained from the Eagon–Northcott resolu-tion. We describe it in details here. Take arbitrary increasing column indices c < c < · · · < c r < c r +1 and arbitrary k with 1 ≤ k ≤ r . We will consider the determinant of the( r + 1) × ( r + 1) matrix A pictured in Figure 3, where the k -th row and the ( k + 1)-throw are identical.By expanding along the k -th row, we arrive at the relation r +1 (cid:88) j =1 ( − j +1 x c j +( k − d det( A c ,..., ˆ c j ,...,c r +1 ) = 0 , where A c ,..., ˆ c j ,...,c r +1 is obtained from the matrix A by removing the k -th row and the j -th column.It is clear that the leading monomial of x c j +( k − d det( A c ,..., ˆ c j ,...,c r +1 ) is x c j +( k − d x c x c + d · · · x c j − +( j − d x c j +1 +( j − d · · · x c r +1 +( r − d , (13) = x c x c k x c k +1 x c r +1 x c + d x c k + d x c k +1 + d x c r +1 + d x c +( k − d x c k +( k − d x c k +1 +( k − d x c r +1 +( k − d x c +( k − d x c k +( k − d x c k +1 +( k − d x c r +1 +( k − d x c + kd x c k + kd x c k +1 + kd x c r +1 + kd x c +( r − d x c k +( r − d x c k +1 +( r − d x c r +1 +( r − d Figure 3.
The matrix A which is a term, up to sign, of the full expansion of det( A ), before cancellation.On the other hand, as the indices of the entries are non-decreasing along the rows andcolumns of A , the lexicographic order guarantees that the leading monomial in the fullexpansion is the diagonal one: x c x c + d · · · x c k +( k − d x c k +1 +( k − d · · · x c r +1 +( r − d . It is precisely when we choose j = k or j = k + 1 in (13). This corresponds to thebinomial in item (ii) given by Nam, after making( c , c + d, . . . , c k +( k − d, c k +1 +( k − d, . . . , c r +1 +( r − d ) = ( α , . . . , α k − , t, α k , . . . , α r ) . With Corollary 3.8 in mind, the last step is to apply the proof of [33, Theorem 5.13]and [8, Proposition 1.1 and Corollares 2.1, 2.2]. (cid:3)
We end this section with some quick applications, since we know the Sagbi bases ofthe fiber cone F ( I ) and the Rees algebra R ( I ). Proposition 3.10. (a)
The fiber cone F ( I ) is a normal Cohen–Macaulay domain.In particular, F ( I ) has rational singularities if char( K ) = 0 , and it is F-rationalif char( K ) > . (b) When r ≥ and d ≥ , the fiber cone F ( I ) is Gorenstein if and only if c ∈ { r, r + 1 , r + d, r + d + 1 , r + d } . (c) The analytic spread of the ideal I , namely, the dimension of the fiber cone is dim( F ( I )) = c + ( r − d, if r + d < c,rc − r + 1 , if r < c ≤ r + d, , if r = c. (14) Proof.
Nam in [33, Theorem 3.28 and Proposition 5.12] showed that R ( I ) is normal,and in( R ( I )) = R (in( I )). Now, F (in( I )) inherits the normality from R (in( I )) by [40,Theorem 7.1]. The deformation theory of the Sagbi basis then gives the normality, Cohen–Macaulayness, and singularity of F ( I ) by Theorem 3.7 and [8, Corollary 2.3]. Andthe Gorensteinness of the fiber cone is the by-products of the Sagbi basis theory with[10, Corollary 2.7] and [16, Theorem 3.7].It remains to prove the item (c). The case when c ≥ r + d follows directly from[16, Theorems 2.3]. And the case when c = r is due to the fact that I is principal. As forthe remaining case when 1 < r < c < r + d , by the reduction before [16, Theorem 2.3],we can reduce the ideal I r ( H r,c,d ) ⊂ R = K [ x ] to some I r ( H r,c,d (cid:48) ) ⊂ K [ x (cid:48) ] with d (cid:48) = c − r .Here, the collection of variables x (cid:48) is a subset of the original collection of variables x , and r ( H r,c,d (cid:48) ) K [ x ] = I r ( H r,c,d ). Whence, c = r + d (cid:48) and by the result in the c = r + d case,the current dimension is rd (cid:48) + 1 = rc − r + 1. (cid:3) Regularity of the fiber cone
In this section, we focus on the Castelnuovo–Mumford regularity of the fiber cone F ( I ) ∼ = K [ I ] for the ideal I = I r ( H r,c,d ) ⊂ R = K [ x , . . . , x c +( r − d ] where 2 ≤ r ≤ c . Wewill fix N = N ( r, c, d ) := dim( R ) = c + ( r − d in the rest of this paper. Recall that we have the natural epimorphisms K [ Y ] = K [ Y α | α ∈ Λ r,d ( N )] (cid:16) K [ I ] = K [ M ( α ) | α ∈ Λ r,d ( N )] , (15)and K [ Y α | α ∈ Λ r,d ( N )] (cid:16) K [in( I )] = K [in( M ( α )) | α ∈ Λ r,d ( N )] . (16)Furthermore, if β i,j is the graded Betti number of K [ I ] considered as a K [ Y ]-module,then the Castelnuovo–Mumford regularity of K [ I ] is defined to bereg( K [ I ]) := max i,j { j − i | β i,j (cid:54) = 0 } . The following main result of this section computes this regularity explicitly.
Theorem 4.1.
The Castelnuovo–Mumford regularities of the fiber cones F ( I r ( H r,c,d )) and F (in( I r ( H r,c,d ))) are given by reg( F ( I r ( H r,c,d ))) = reg( F (in( I r ( H r,c,d )))) = N − − (cid:98) ( N − /r (cid:99) , if r + d ≤ c,dr − r − d + 2 c − , if r + d < c < r + d, ( r − c − r − , if r < c ≤ r + d, , if r = c. Proof.
Let P and P (cid:48) be the kernels of the epimorphisms in (15) and (16) respectively. ByCorollary 3.8, in( P ) := in τ ( P ) = (cid:104) Y α Y β | ( x α > lex x β ) is unsorted (cid:105) is squarefree. Meanwhile, by Theorem 3.7 and [8, Corollary 2.2], there exists some termorder τ (cid:48) such that in τ ( P ) = in τ (cid:48) ( P (cid:48) ). Then, by our favorite [10, Corollary 2.7], we willhave reg( K [ Y ] /P ) = reg( K [ Y ] / in( P )) = reg( K [ Y ] / in τ (cid:48) ( P (cid:48) )) = reg( K [ Y ] /P (cid:48) ) . Therefore, it amounts to explore the regularity of K [ Y ] / in( P ) considered as a K [ Y ]-module.Note that F ( I ) = K [ I ] ∼ = K [ Y ] /P is Cohen–Macaulay by Proposition 3.10 (a). Con-sequently, by [10, Corollary 2.7], K [ Y ] / in( P ) is Cohen–Macaulay as well.Now, back to the probe of the regularity. The extremal situation when r = c is easysince the ideal is principal. Meanwhile, the case when r < c < r + d can be reduced to thecase c = r + d by the reduction stated before [16, Theorem 2.3]; see also the explanationin the proof of Proposition 3.10. Now, we can focus on the cases when c ≥ r + d . Whence,by applying Proposition 3.10 (c), the Auslander–Buchsbaum theorem [5, Theorem 1.3.3]and the Eagon–Reiner theorem [25, Theorem 8.1.9], we know that the Alexander dual ofthe squarefree ideal in( P ) has a linear resolution with regularity (cid:102) N − N or (cid:102) N − ( rd + 1)when c > r + d or c = r + d respectively. Here, (cid:102) N = dim( K [ Y ]), namely the cardinality ofΛ r,d ( N ). Our task is then to find the projective dimension of (in( P )) ∨ , by [25, Proposition .1.10]. This task can be completed by combining the coming Propositions 4.15, 4.16 and4.19. (cid:3) Here is the plan on how to complete the final task stated just now. The concretecomputation of pd((in( P )) ∨ ) will be carried out for the case 2 r + d ≤ c in the subsection4.4, for the case r + d < c < r + d in the subsection 4.5 and for the case r ≤ c ≤ r + d in thesubsection 4.6. However, before that, we need to describe explicitly the minimal monomialgenerators of (in( P )) ∨ in subsection 4.1. We will give a linear order on the minimalmonomial generating set G ((in( P )) ∨ ), and show that (in( P )) ∨ has complete intersectionquotients in the subsection 4.2. The maximal length of the complete intersections ingeneral only gives an upper bound of the projective dimension. By studying the iteratedmapping cones carefully, we will show that the projective dimension is actually achievedby the maximal length. The general strategy will be stated in the subsection 4.3. And itis one of the essential parts for the computations in the subsections 4.4, 4.5 and 4.6.4.1. Minimal monomial generators of (in( P )) ∨ . In the following, we shall deal withtwo different linear order > on the variables of K [ Y ] = K [ Y α | α ∈ Λ r,d ( N )]:(i) lex type : Y α > Y β ⇔ x α > lex x β , or(ii) revlex type : Y α > Y β ⇔ x α > revlex x β . (17)We will then consider the lexicographic order > lex on K [ Y ] with respect to either givenlinear order > on the variables.It is already known that a minimal monomial generating set of in( P ) = in τ ( P ) is givenby { Y α Y β | x α > lex x β but ( x α , x β ) is not sorted } . It is not difficult to see that this is also { Y α Y β | x α > revlex x β but ( x α , x β ) is not sorted } . Let G be the simple graph on the vertex set { Y α | α ∈ Λ r,d ( N ) } , such that { Y α > Y β } is an edge if and only if ( x α , x β ) is sorted . It is then clear that in( P ) is the edge ideal ofthe complement graph G (cid:123) .In the remaining of this subsection, we will only consider the case when c > r + d ; thecase when r ≤ c ≤ r + d , which is only slightly different, will be left in the final subsection4.6. Whence, as the ideal (in( P )) ∨ has a linear resolution by the proof of Theorem 4.1,it is minimally generated by some squarefree monomials of degreereg((in( P )) ∨ ) = (cid:102) N − N by (14).Suppose that Y α Y α · · · Y α (cid:101) N − N ∈ G (in( P ) ∨ ) is a minimal monomial generator. This isequivalent to saying that (cid:110) Y α , . . . , Y α (cid:101) N − N (cid:111) is a minimal vertex cover of the complementgraph G (cid:123) , by [25, Corollary 9.1.5]. In other words, this means that { Y β , . . . , Y β N } := (cid:110) Y α , . . . , Y α (cid:101) N − N (cid:111) (cid:123) is a maximal clique of G . Notice that the cardinality of this set is precisely N . Notation 4.2. (1) For any squarefree monomial u ∈ K [ Y ], write (cid:98) u := Ñ (cid:89) α ∈ Λ r,d ( N ) Y α é (cid:44) u. n the other hand, if F is a subset of the vertex set of G , we will write Y F := (cid:89) f ∈ F f. (2) For any increasing sequence β = ( b < b < · · · < b s ), we will write β k = b k for k = 1 , , . . . , s . We will also use β ≤ k for the subsequence ( b < b < · · · < b k ) andsimilarly define β ≥ k . Meanwhile, β { k ,...,k } will be the subsequence ( b k < b k +1 < · · · < b k ).Therefore, the minimal monomial generating set of in( P ) ∨ is given by (cid:110) ‘ Y F (cid:12)(cid:12)(cid:12) F is a maximal clique of G (cid:111) . (18)Next, we describe the maximal cliques of G . Lemma 4.3.
The set F = { Y β > · · · > Y β N } (19) is a maximal clique of G if and only if F satisfies the following three groups of conditions: ≤ β ≤ β ≤ · · · ≤ β N ≤ β ≤ β ≤ · · · ≤ β N ≤ · · · ≤ β r ≤ β r ≤ · · · ≤ β rN ≤ N ; (20) β j k k + 1 = β j k k +1 for some j k , while β jk = β jk +1 for j (cid:54) = j k ; (21) β = 1 , β ≤ r − N = β ≥ and β rN = N. (22) Proof. If F is a maximal clique of G , then the pair ( x β i , x β j ) is sorted whenever 1 ≤ i 1, there exists a unique j k such that the condition (21) is met.Consequently, we have N − r (cid:88) j =1 ( β jN − β j ) = r − (cid:88) j =1 ( β jN − β j ) + ( β rN − β r ) ≤ r − (cid:88) j =1 ( β j +11 − β j ) + ( N − β r ) = N − β ≤ N − . Therefore, we obtain the condition (22).On the other hand, it is not difficult to see that the conditions (20), (21) and (22)together are sufficient for the set F in (19) to be a maximal clique of G . (cid:3) The following remark gives us a better view of the maximal clique with respect to therunning indices. Remark 4.4. Notice that given the initial sequence β , the maximal clique F in (19) iscompletely determined by the moving sequence : λ F = ( j , j , . . . , j N − ) , (23)where the j k ’s are given by the condition (21). We will call the integer j k as the movement from Y β k to Y β k +1 and call it a j k -movement in λ F .On the other hand, notice that the cardinalities |{ k | j k = t }| = β t +11 − β t for t = 1 , , . . . , r − 1, (24)and |{ k | j k = r }| = N − β r , (25)by the condition (22). Since β = 1, the moving sequence λ F in (23) completely de-termines the sequence β , and consequently, determines the maximal clique F itself. Inshort, knowing the maximal clique F is equivalent to knowing the moving sequence λ F . .2. Complete intersection quotients. Next, we consider the following total order (cid:31) on the minimal monomial generating set of (in( P )) ∨ : ’ Y F (cid:31) ’ Y F ⇔ Y F > lex Y F . We will also write it correspondingly as F (cid:31) F . The very first maximal clique of G will be denoted sometimes by F − . As a reminder, we have two different orderings onthe variables. Consequently, we have two different strategies for this ordering. They aretailored to the different cases studied later.In this subsection, the plan is to show, for each maximal clique F as in (19), the colonideal I F := (cid:104) ‘ Y F (cid:48) | F (cid:48) is also a maximal clique of G such that F (cid:48) (cid:31) F (cid:105) : ‘ Y F (26)is a complete intersection. Again, we will only consider the case when c > r + d ; thecase when r ≤ c ≤ r + d will be left in the final subsection. Now, take arbitrary Y H inthe minimal monomial generating set of the colon ideal I F . Therefore, there exists some F (cid:48) (cid:31) F such that (cid:104) ‘ Y F (cid:48) (cid:105) : ‘ Y F = (cid:104) Y H (cid:105) . Whence, H = F \ F (cid:48) . Suppose for later reference that F (cid:48) = { Y α > · · · > Y α N } . (27)The minimal monomial generating set of I F consists of two types of squarefree monomials,and we will discuss in detail. Observation 4.5 ( Tail generators ) . Suppose that α (cid:54) = β . Since Y F (cid:48) (cid:31) Y F , Y α >Y β and consequently Y β is not the very first ring variable Y , d, d,...,r +( r − d of K [ Y ].For each j = 1 , , . . . , r − 1, we introduce the invariant δ j := min (cid:110) i (cid:12)(cid:12)(cid:12) β { j,...,r − } i = β { j,...,r − } N (cid:111) (28)for partially measuring how far the variables in F are from the last one. It is clear that δ ≥ δ ≥ · · · ≥ δ r − > k ≤ r − < d d < d d · · · < d k + ( k − d < d β kδ k − < d β k +1 δ k − < d · · · < d β r − δ k − . (29)Notice that β r − δ r − − + 1 = β r − δ r − = β r − N = β r and Y β is not the first variable of thering. Thus, β r − δ r − − ≥ β r − ≥ r + ( r − d . Furthermore, note that the two underlinedparts hold automatically. Thus, the condition in (29) always holds for k = r − k ≤ r − k + ( k − d < d β kδ k − , i.e., k + kd < β kδ k − . (30)For each j ≤ r − 1, notice that β jδ j − ≥ β jδ j − β jN − β j +11 − ≥ j + 1 + ( j + 1 − d − j + jd. Thus, by the minimality of the chosen k , one actually has β j = j + ( j − d for j = 1 , , . . . , k . (31)Now, for the fixed k introduced above, we call the sequence in (29) as γ . It is clearthat β ≤ k = γ ≤ k by (31). Meanwhile, for j = k + 1 , . . . , r , one has γ j = β j − δ k − ≤ β j − N = β j . hus, β ≥ γ componentwise. Meanwhile, by the choice of δ k , β { k,...,r − } δ k − (cid:54) = β { k,...,r − } N .Consequently Y β < Y γ in both cases in (17). Now, one can easily construct some maximalclique F (cid:48)(cid:48) of G as F (cid:48)(cid:48) = ® Y γ = γ > Y γ > · · · > Y γ k (cid:48)− > Y γ k (cid:48) = β > Y γ k (cid:48) +1 = β > · · · >Y γ k (cid:48) + δk − = β δk − > Y γ k (cid:48) + δk − > · · · > Y γ N ´ , using Lemma 4.3. Here k (cid:48) − (cid:80) rj =1 β j − (cid:80) rj =1 γ j = (cid:80) rj = k +1 ( β j − γ j ). And it is clearthat F \ F (cid:48)(cid:48) = ¶ Y β δk > · · · > Y β N © . Now, back to the maximal clique F (cid:48) (cid:31) F with α (cid:54) = β . We claim that F \ F (cid:48)(cid:48) ⊆ F \ F (cid:48) .Otherwise, there exists some i with δ k ≤ i ≤ N such that Y β i ∈ F (cid:48) . Notice that β ≤ k ≤ α ≤ k componentwise by the condition (31). Meanwhile, β ≥ k +11 = β { k,...,r − } N = β { k,...,r − } i comp. ≤ α { k,...,r − } N = α ≥ k +11 . Here, the equalities are due to the choice of δ k and the condition (22), while the com-ponentwise inequality is due to the condition (20) and the fact that Y β i ∈ F (cid:48) . Thus, wehave α ≥ β componentwise. Consequently, Y α ≤ Y β in term of the lexicographicalorder that we define in (17), which is a contradiction. This confirms our claim.Now, by the minimality of H , we must have H = { Y β δk > · · · > Y β N } . We will call themonomial Y H as the tail generator of the corresponding colon ideal I F . Observation 4.6 ( Corner generators ) . Suppose that α = β , whence α N = β N bythe condition (22). Furthermore, if λ F and λ F (cid:48) are the corresponding moving sequencesof F and F (cid:48) respectively, then λ F and λ F (cid:48) coincide if we deem them as multi-sets by theconditions (24) and (25).Depending on the linear order of the variables given at (17), we have two cases.(i) The ordering is of lex type. In this case, if we also consider λ F and λ F (cid:48) astuples in Z r − , then F (cid:48) (cid:31) F if and only ifthe leftmost nonzero component of the difference λ F (cid:48) − λ F is positive. (32)Now consider the set K F := { ≤ k ≤ N − | j k < j k +1 } (33)where the j k ’s are given in (21). For each fixed k ∈ K F , we then have somemaximal clique F (cid:48)(cid:48) with the moving sequence λ F (cid:48)(cid:48) = ( j , . . . , j k − , j k +1 , j k , j k +2 , . . . , j N − ) . Here, one uses the condition (20) to verify the existence (compatibility) of F (cid:48)(cid:48) . Itis clear that F (cid:48)(cid:48) (cid:31) F and F \ F (cid:48)(cid:48) = { Y β k +1 } .It remains to show that a minimal monomial generator of the colon ideal I F comes from this situation if and only if it belongs to ¶ Y β k +1 | k ∈ K F © . (34)Notice that the “if” part has been shown by the above argument. And the indicesin K F cuts λ F into a collection of weakly decreasing subsequences.As a quick example, when ( r, c, d ) = (3 , , F = ß Y , , j −→ Y , , j −→ Y , , j −→ Y , , j −→ Y , , j −→ Y , , j −→ Y , , j −→ Y , , ™ . he moving sequence is λ F = ( j , j , . . . , j ) = (3 , , , , , , . Whence, K F = { } and the expected generator in this situation is solely Y , , .And the index 3 in K F cuts λ F into the underlined format.Now, back to the argument for F (cid:48) in (27) such that the corresponding mono-mial Y H is a minimal monomial generator of the colon ideal I F . Let A = { ≤ k ≤ N − | α k +1 (cid:54) = β k +1 } . It remains to show that A ∩ K F (cid:54) = ∅ due tothe expected minimal generating set (34) and the fact that H = F \ F (cid:48) . Noticethat the maximal cliques F and F (cid:48) surely contain ¶ Y β k +1 | k / ∈ A © ∪ { Y β , Y β N } .Now, suppose for contradiction that A ∩ K F = ∅ . Consequently, both F and F (cid:48) contain ¶ Y β k +1 | k ∈ K F © ∪ { Y β , Y β N } = ¶ Y β j > Y β j > · · · > Y β jt © . It is clear that j = 1 and j t = N . Furthermore, for each s ∈ [ t ], we have β j s = α j s . For each s ∈ [ t − λ s is the sequence of movement in F from β j s to β j s +1 and similarly introduce λ (cid:48) s for F (cid:48) . It is clear that λ s is a weaklydecreasing sequence, and |{ r (cid:48) -movements in λ s }| = β r (cid:48) j s +1 − β r (cid:48) j s = |{ r (cid:48) -movements in λ (cid:48) s }| for every r (cid:48) ∈ [ r ]. Thus, F (cid:31) F (cid:48) by the characterization in (32), which contradictsthe assumption that F (cid:48) (cid:31) F .In short, in this lex type ordering case with the assumption that α = β , themonomial Y H is a minimal generator of I F if and only if this monomial belongsto the set in (34).(ii) The ordering is of revlex type. In this case, if we also consider λ F and λ F (cid:48) astuples in Z r − , then F (cid:48) (cid:31) F if and only if the leftmost nonzero component of thedifference λ F (cid:48) − λ F is negative .Now, let K (cid:48) F be the set of all indices k such that j k > j k +1 with the additionalrequirement that we have some maximal clique F (cid:48)(cid:48) with λ F (cid:48)(cid:48) = ( j , . . . , j k − , j k +1 , j k , j k +2 , . . . , j N − ) . Notice that, unlike the lex type case, such a maximal clique F (cid:48)(cid:48) does not existby default. Now, for any such k in K (cid:48) F with the accompanied F (cid:48)(cid:48) , it is clear that F (cid:48)(cid:48) (cid:31) F and F \ F (cid:48)(cid:48) = { Y β k +1 } . One can argue as in the lex type case that theminimal monomial generators coming from this situation is purely ¶ Y β k +1 | k ∈ K (cid:48) F © . (35)Two maximal cliques in the case ( r, c, d ) = (2 , , 1) are pictured in Figure 4. The lineargenerators generated by the current method are marked with ∗ . For the obvious pictorialreasons, we will call the minimal generators given by either (34) or (35), depending onthe ordering of the variables, as the corner generators of the corresponding colon ideal I F . Remark 4.7. As a reminder, the tail generator may or may not belong to the minimalmonomial generating set of I F , depending on whether it will be canceled out by somecorner generator. For instance, when ( r, c, d ) = (2 , , 1) we will have a maximal clique F = { Y , > Y , > Y , > Y , > Y , > Y , > Y , > Y , > Y , } ; , Y , Y , Y , ∗ Y , Y , ∗ Y , Y , Y , (cid:77) Y , Y , Y , Y , ∗ Y , Y , ∗ Y , Y , ∗ , (cid:77) Y , (cid:77) Figure 4. Two maximal cliques for ( r, c, d ) = (2 , , 1) in the lex typeordering. Variables in the respective tail generators are marked with (cid:77) and corner generators are marked with ∗ .see also the right-hand side picture in Figure 4. Using the lex type ordering in (17), wewill have a tail generator Y H = Y , Y , regarding this F . But it is not difficult to see that the set of corner generators is precisely { Y , , Y , , Y , } . Whence, the tail generator is canceled out and the colon ideal I F = (cid:104) Y , , Y , , Y , (cid:105) .On the other hand, all the corner generators will stay for sure, since they are linear.Whence, the colon ideal I F is always a monomial complete intersection ideal.4.3. Upper bound for the projective dimension. Recall that for a homogeneousideal I and a homogeneous element f , the short exact sequence0 → R/ ( I : f ) → R/I → R/ ( I + f ) → R/ ( I + f )) ≤ max { pd( R/ ( I : f )) + 1 , pd( R/I ) } . Whence, if I = (cid:104) f , . . . , f s (cid:105) such that for each t = 2 , , . . . , s , the colon ideal (cid:104) f , . . . , f t − (cid:105) : f t is a complete intersection of codimension (cid:96) t , thenpd( I ) = pd( R/I ) − ≤ max ≤ t ≤ s (cid:96) t . (36)By the argument in the previous subsection, we see that the minimal monomial gen-erating set of the colon ideal I F corresponding to the maximal clique F is given by(a) some corner generators, which are all linear, and(b) a unique tail generator, if it exists and is not canceled out by corner generators.Furthermore, notice that if the starting variable of F is the first variable of the ring, thetail generator does not exist.In this subsection, we intend to give an upper bound of the projective dimension of(in( P )) ∨ . Since the tail generator, if exists, only contributes 1 to the codimension ofthe complete intersection ideal I F , roughly speaking, it suffices to figure out when themaximal number of the corner generators is achieved. Observation 4.8. In the following, we assume that the maximal number of the cornersis achieved for some maximal clique F = { Y β > · · · > Y β N } . We will only explain in thecase of lex type ordering of variables, which is sufficient for obtaining the upper bound in λ F is a concatenation of some strictly increasingsubsequences: λ F = ( λ , λ , . . . , λ s ) . Here, we always require that the strictly increasing subsequences above to be maximalwith respect to their individual lengths. For instance, for the example in Remark 4.7, themoving sequence is λ F = (3 , , , , , , , , where the maximally strictly increasing subsequences are underlined. In this special case,only λ = (2 , 3) has length 2; all other subsequences have only length 1.Now, back to our moving sequence λ F , we observe that the two sets { ≤ i ≤ N − | i − / ∈ K F } and { the position of the variable in F after the movements of λ j | j = 1 , , . . . , s − } are identical, where the set K F is defined in (33). Therefore, maximizing the cardinalityof K F is equivalent to minimizing the number of strictly increasing subsequences.Consequently, we obtain the following main result of this subsection. Proposition 4.9. Under the lex type or revlex type ordering of the variables, let (cid:96) F bethe codimension of the complete intersection colon ideal I F corresponding to the maximalclique F studied above. Then, pd((in( P )) ∨ ) ≤ max F (cid:54) = F − (cid:96) F , (37) where F − is the first maximal clique of G . In particular, when c > r + d , we have pd((in( P )) ∨ ) ≤ ( N − − (cid:100) ( N − /r (cid:101) + 1 . Proof. The first inequality is from (36). With Observation 4.8, we notice that each strictlyincreasing subsequence has a length at most r . Therefore, | K F | ≤ N − − (cid:100) ( N − /r (cid:101) , (38)since N − | F | − λ F . Note that the final +1 of the “in particular”part comes from the potential contribution of the tail generator. (cid:3) Remark 4.10. When applying the revlex type of ordering, by symmetry, we will breakthe moving sequence λ F into maximally strictly decreasing subsequence, again, writtenas λ F = ( λ , λ , . . . , λ s ). The final movement of each λ j will not contribute any cornergenerator. Furthermore, some additional movement may fail to contribute, due to themodified description of K (cid:48) F in Observation 4.6. For instance, when ( r, c, d ) = (3 , , F = { Y , , > Y , , > Y , , > Y , , > Y , , > Y , , > Y , , > Y , , } . The associated moving sequence is λ F = (3 , , , , , , Y , , is the onlycorner generator.Actually, we seek more than a mere upper bound. To get over this discrepancy, weneed extra tools. Recall the following key facts of iterated mapping cones from [36,Construction 27.3]. Let I be an ideal minimally generated by monomials m , . . . , m r insome polynomial ring S . For 1 ≤ i ≤ r , write I i := (cid:104) m , . . . , m i (cid:105) . Then we have a shortexact sequence of homogeneous modules0 → ( S/ ( I i : m i +1 ))( m i +1 ) m i +1 → S/I i → S/I i +1 → , here the comparison map is the multiplication by m i +1 . Here, ( S/ ( I i : m i +1 ))( m i +1 )is obtained by shifting the module S/ ( I i : m i +1 ) in multidegree by m i +1 so that thecomparison map is of degree 0. Assume that a multigraded free resolution F i of S/I i isalready known and that a multigraded free resolution G i of S/ ( I i : m i +1 ) is also known,then one can construct the mapping cone and obtain a multigraded free resolution F i +1 of S/I i +1 .Now, let us focus on the complete intersection quotients studied above in the polynomialring S = K [ Y ]. For each given maximal clique F which is different from the very firstone F − , the colon ideal is a complete intersection ideal I F . Let G F := G ( I F ) be theminimal monomial generating set of the monomial ideal I F . Whence, it can be resolvedminimally by the Koszul complex, whose top module has rank 1 at the homological degree (cid:96) F = | G F | , and is shifted in multidegree by m F := (cid:81) f ∈ G F f . This module is denoted by S ( m F ) in [36]. Similarly, the free module at the homological degree (cid:96) F − (cid:76) f ∈ G F S ( m F /f ). Definition 4.11. For each maximal clique F studied above, we define the essential part of F as Ess( F ) := { g ∈ F | for each f ∈ G F , g (cid:54) | f } = F \ Supp( m F ) , where Supp( m F ) denotes the set of variables that divide the monomial m F .Since the monomial generator of (in( P )) ∨ that we are dealing with is ‘ Y F , in themapping cone , we have to further shift in multidegree by ‘ Y F . Consequently, we will havea rank 1 module shifted in multidegree by m F ‘ Y F = ÿ(cid:0) Y Ess( F ) at the homological degree (cid:96) F + 1. Meanwhile, the free module at the homological degree (cid:96) F of the shifted Koszulcomplex is (cid:77) f ∈ G F S ( m F ‘ Y F /f ) = (cid:77) f ∈ G F S ( ¤(cid:0) Y Ess( F ) (cid:116) Supp( f ) ) . Note that the free resolution given by the iterated mapping cone may not necessarilybe minimal. For instance, when ( r, c, d ) = (3 , , 1) and if we apply the lex type orderingof the variables, then pd((in( P )) ∨ ) = 4 < max F (cid:54) = F − (cid:96) F = 5 . This is also the reason why we need to consider the revlex type of ordering in some cases.We need the following key observation to determine the regularity of the fiber cone. Lemma 4.12. Suppose that there is some maximal clique F satisfying the following threeconditions: (C1) (cid:96) F = max F (cid:54) = F − (cid:96) F , (C2) there is no other maximal clique F (cid:48) with Ess( F (cid:48) ) = Ess( F ) , and (C3) there is no maximal clique F (cid:48) and f ∈ G F (cid:48) with the disjoint union Ess( F (cid:48) ) (cid:116) Supp( f ) = Ess( F ) .Then, the projective dimension is given by pd((in( P )) ∨ ) = max F (cid:54) = F − (cid:96) F . Proof. The condition (C1) is obviously necessary in view of (37). Now, we go throughthe iterated mapping cone process.When doing the mapping cone at the stage of the maximal clique F = F , the toprank 1 module is now at the homological degree (cid:96) F + 1. It does not contribute to theprojective dimension as expected, precisely when it actually cancels some free module ofrank 1 at the homological degree (cid:96) F , shifted at the same multidegree. By the maximality f (cid:96) F , the latter free module has two sources. The first source is the top free module ofsome Koszul complex added earlier by some maximal clique F (cid:48) with (cid:96) F (cid:48) = (cid:96) F − 1. Sincewe will have (cid:96) F ≥ F (cid:48) (cid:54) = F − . This leads to theintroduction of the condition (C2). The second source is the free module from the secondhighest homological degree in earlier Koszul complexes. This leads to the introduction ofthe condition (C3).If the top rank 1 module “survives” at this stage, it will “survive forever”. This isbecause it stays at the homological degree (cid:96) F + 1, while all existing free modules that arecanceled out in later mapping cones, undoubtedly come from the homological degrees atmost (cid:96) F . (cid:3) Remark 4.13. (1) Suppose that we have a maximal clique F (cid:48) and f ∈ G F (cid:48) withthe disjoint union Ess( F (cid:48) ) (cid:116) Supp( f ) = Ess( F ). Since the starting variable of F belongs to Ess( F ) and the starting variable of F (cid:48) belongs Ess( F (cid:48) ), these twostarting variables must coincide. Furthermore, f cannot be this common variable.(2) Similarly, if we have two maximal cliques F (cid:48) and F with Ess( F (cid:48) ) = Ess( F ), thenthe starting variable of F (cid:48) is identical to that of F .4.4. The r + d ≤ c case. Starting from this subsection, we will pin down the concreteprojective dimension of (in( P )) ∨ . For that purpose, suppose that N − pr + q suchthat 1 ≤ q ≤ r . To achieve the maximum in (38), the most natural strategy in mind isto assume that λ F is the concatenation of p subsequences of the form (1 , , . . . , r ) first,and then followed by 1 subsequence of the form (1 , , . . . , q ).We have seen earlier that the moving sequence λ F in (23) actually determines themaximal clique F . Using the above strategy, it is not difficult to see that the firstvariable of F has the form β = (1 , p +1) , p +1) , . . . , q ( p +1) , q ( p +1)+ p, . . . , q ( p +1)+( r − q − p )by (24) and (25). Since we have the < d compatibility requirement, to make the movingsequence legal, we have the following considerations.(i) If r − q − ≥ 1, i.e., if q ≤ r − 2, then the compatibility requirements are reducedto the single inequality 1 + q ( p + 1) + 1 < d q ( p + 1) + p , which is equivalentto saying c ≥ r + d + 2.(ii) If q = r − 1, then the compatibility requirements are reduced to the single in-equality 1 + 1 < d p + 1), which is equivalent to saying c ≥ r + d .(iii) If q = r , then the compatibility requirements are reduced again to the singleinequality 1 + 1 < d p + 1), which is equivalent to saying c ≥ r + d + 1.Notice that if c = 2 r + d , then N − d + 1) r + ( r − q = r − 1. And if c = 2 r + d + 1, then N − d + 1) r + r . Whence, q = r . Now, it suffices to considerthe following two cases. Observations 4.14. (a) If c ≥ r + d and q < r , then the above argument showsthat we can find suitable maximal clique F , whose moving sequence λ F has p + 1strictly increasing subsequences, and the final movement in the final subsequenceis q . Since 1 + ( p + 1) > d in these cases, the special k in (29) is actually 1, by(31). Now, since q < r , we have a linear tail generator by the description in thesubsection 4.2. This generator surely cannot be canceled by a corner generator.Hence the maximum at the right-hand side of (37) is exactly( N − − (cid:100) ( N − /r (cid:101) + 1 = ( N − − (cid:98) ( N − /r (cid:99) . b) If c ≥ r + d + 1 and q = r , suppose that the moving sequence λ F of some suitablemaximal clique F has the least number of strictly increasing subsequences. Since q = r , this forces λ F to be exactly the concatenation of p + 1 subsequences of theform (1 , , . . . , r ). Whence, the special k can be seen to still be 1, while the tailgenerator has degree 2. This tail generator will be canceled by the final cornergenerator Y β N − . In other words, the minimal monomial generating set of I F consists solely of corner generators. Since r divides N − 1, the maximum at theright-hand side of (37) is then ( N − − ( N − /r .Here is the main result of this subsection. Proposition 4.15. If r + d ≤ c , then pd((in( P )) ∨ ) = N − − (cid:98) ( N − /r (cid:99) .Proof. Let F be the maximal clique given in Observations 4.14. We will verify that F satisfies the conditions (C2) and (C3) in Lemma 4.12, where (C1) holds already.Suppose that F (cid:48) is also a maximal clique with Ess( F ) = Ess( F (cid:48) ). Without loss ofgenerality, we may assume that F = { Y β > · · · > Y β N } while F (cid:48) = { Y α > · · · > Y α N } .(i) We have seen in Remark 4.13 that β = α .(ii) Notice that Y β r , Y β r , . . . , Y β pr are the next p elements in order in Ess( F ).Therefore, Y α s r + s are all corner generators of I F (cid:48) for 0 ≤ s ≤ p − ≤ s ≤ r − 1. But in any legitimate moving sequence, any strictly increasingsubsequence has length at most r . Hence, the movement from Y α to Y α pr in F (cid:48) consists precisely of the concatenation of p subsequences of the form (1 , , . . . , r ),just as in F . Thus, as β = α , it forces α j = β j for j = 1 , , . . . , pr .(iii) For the final moving subsequence, we have two cases. Note that since α = β ,the special k in (29) for F (cid:48) is actually 1 as well, by (31).(a) Suppose that q < r . Whence, by the discussion in Observations 4.14, wehave no further variable for Ess( F ). Since α N = β N , the remaining movingsubsequence of F (cid:48) is just a rearrangement of (1 , , , . . . , q ) while q < r .Whence, the tail generator corresponding to F (cid:48) is linear as well. In turn, Y α pr +1 , . . . , Y α N − are all corner generators for F (cid:48) , since Ess( F ) = Ess( F (cid:48) ).Now, the moving subsequence corresponding to the movement from Y α pr to Y α N has to be a strictly increasing one, namely (1 , , . . . , q ). In short, λ F = λ F (cid:48) . Consequently, we have F = F (cid:48) , as expected.(b) Suppose that q = r . Whence, by the discussion in Observations 4.14, Y β N is the unique additional variable for Ess( F ). Since Ess( F ) = Ess( F (cid:48) ) whilethe final variable of F (cid:48) can never be a corner generator, this final variablewill belong to a tail generator of degree at least 2. On the other hand, sincethe remaining moving subsequence contains exactly one copy of r , this tailgenerator has degree at most 2. This implies that Y α pr +1 , . . . , Y α N − are allcorner generators regarding F (cid:48) , and Y α N − will cancel the tail generator ofdegree 2. As in the previous case, we will consequently have F = F (cid:48) , asexpected.So far, the conditions (C1) and (C2) have been verified. We continue to verify thecondition (C3) in these cases. Assume to the contrary that there is some maximal clique F (cid:48) and some generator f ∈ G F (cid:48) with the disjoint union Ess( F (cid:48) ) (cid:116) Supp( f ) = Ess( F ). ByRemark 4.13, F and F (cid:48) have the same starting variable, which cannot be f .(a) When c ≥ r + d and q < r , we have seen thatEss( F ) = ¶ Y β , Y β r , Y β r , . . . , Y β pr © . ince Supp( f ) ⊂ Ess( F ) and f ∈ G F (cid:48) , this f has to be a corner generator in G F (cid:48) and is linear.If f = Y β p (cid:48) r with 1 ≤ p (cid:48) < p , then in the moving sequence λ F (cid:48) , there will be astrictly increasing subsequence of length 2 r moving from Y β p (cid:48)− r to Y β p (cid:48) +1) r .But the length of a strictly increasing subsequence is at most r , and we have acontradiction.If f = Y β pr , then either we have a strictly increasing subsequence of length > r , or this f belongs to the tail generator. The first case won’t happen by thesame reason. As for the latter one, notice that the special k in (29) has to be 1.This implies that starting from f all the movements in F (cid:48) have to be r . But thisis impossible, since β = α and we have the requirements (24) and (25).(b) When c ≥ r + d + 1 and q = r , like above, we haveEss( F ) = ¶ Y β , Y β r , Y β r , . . . , Y β pr , Y β N =1+( p +1) r © . Similarly, we only have to worry about the case when f is the last one, which inthis case is Y β N = Y α N . Consider the strictly increasing subsequence that movestarting from Y α pr = Y β pr regarding F (cid:48) . We claim that the end variable of thisround of movements will be the starting variable of the tail generator of I F (cid:48) . Tosee it, notice that if the tail generator has a higher degree, it will get canceledout by some corner generator in G F (cid:48) . This will make Y α N appearing in Ess( F (cid:48) ),a contradiction. On the other hand, if the tail generator has a lower degree, theend variable of this round will appear in Ess( F (cid:48) ), another contradiction.Notice that the remaining moving subsequence starting form Y α pr is just arearrangement of (1 , , . . . , r ) by the requirements (24) and (25). If the tail gener-ator is linear, this generator is precisely f . Whence λ F (cid:48) = λ F , and consequently F (cid:48) = F , a contradiction. On the other hand, when the tail generator is not linear,since β = α and we have the special k in (29) for F (cid:48) to be 1, this tail generatorhas degree 2 and the final movement is r . Therefore, the strictly increasing mov-ing subsequence starting from Y α pr will be (1 , , . . . , r − λ F (cid:48) = λ F , which is a contradiction. And this completes the proof. (cid:3) The r + d < c < r + d case. The main result of this subsection is to show thefollowing formula. Proposition 4.16. Suppose that r + d < c < r + d . Then the projective dimension isgiven by pd((in( P )) ∨ ) = dr − r − d + 2 c − . (39)Though this is just an intermediate result with respect to our Theorem 4.1, its proofis still quite involved. We will denote the upper bound at the right-hand side of (37) by (cid:96) lexr,c,d or (cid:96) revlexr,c,d , depending on whether we take the lex type or revlex type ordering ofvariables. And here is the strategy for proving (39).(I) We first show that the equality (39) holds when c = r + d + 1 via the lex typeordering. Whence, the right-hand side of (39) is given by (cid:96) lexr,r + d +1 ,d , which surelysatisfies (cid:96) lexr,r + d +1 ,d ≤ (cid:96) revlexr,r + d +1 ,d by (37).(II) Next, we show that (39) holds when c = 2 r + d − revlex type ordering.Whence, the right-hand side of (39) is given by (cid:96) revlexr, r + d − ,d .(III) As for c with r + d + 1 ≤ c < r + d − 1, we show that (cid:96) revlexr,c,d + d ≤ (cid:96) revlexr +1 ,c +1 ,d . Dueto the format of the right-hand side of (39) and the established equalities in the revious two cases, we actually have equality here. Whence, the right-hand sideof (39) is precisely (cid:96) revlexr,c,d whenever r + d < c < r + d .(IV) In the last step, we construct a facet F giving the desired upper bound (cid:96) revlexr,c,d , andshow as in the previous section that the projective dimension in mind is exactlythis bound.4.5.1. Step (I) . In the extremal case when c = r + d + 1, applying the condition in (22)with the < d requirement, it is easy to check that all maximal cliques have the commonstarting variable Y β = Y , d, d,...,r +( r − d , which is the first ring variable. This implies that when calculating the colon ideal I F ,no tail generator will ever appear. Whence, the ideal (in( P )) ∨ has linear quotients, andthe projective dimension of this ideal is precisely the maximal number of the corners by[25, Corollary 8.2.2].Now, we will apply the lex type ordering. The computation of this subcase is basedon the following key observation. Remark 4.17. We have seen earlier in Remark 4.4 that any maximal clique F is deter-mined by its moving sequence λ F . We can break this sequence into maximally strictlyincreasing subsequences λ , . . . , λ s , just as what we did in Observation 4.8.Suppose that when we are processing some subsequence λ t with 1 ≤ t ≤ s , we canmove some variable at the position p with 1 ≤ p ≤ r . But instead, in F , we moveother positions. This implies that this p does not belong to λ t , but rather, belongs tosome λ t (cid:48) with t < t (cid:48) . Let the t (cid:48) be minimal with respect to this property. Then p doesnot appear in any of λ t , . . . , λ t (cid:48) − . One can check that if we move the p to any of the λ t , . . . , λ t (cid:48) − , we still get an allowable moving sequence, i.e., the corresponding maximalclique is legitimate. Notice that after this change, the number of strictly increasingsubsequences does not increase. And it decreases precisely when λ t (cid:48) consists solely of p .The consequence of this observation is that, among all maximal cliques with the samestarting variable, a clique with the minimal number of strictly increasing subsequencescan be obtained by applying the following strategy. Namely, after the previous roundof movement, we scan the movement choices from 1 to r in order. For each choice p ,if the corresponding movement is legitimate (it should satisfy the < d condition and theresulting variable should precede the final variable Y β N ), we adopt this choice into themoving sequence and consider the next choice p + 1. If the movement p is not allowed,then we will consider the next choice p + 1 directly. After we have considered the lastpossible movement r , we call it a round and rewind to consider a new round. If thecurrent β t reaches the final one, we stop.Now, for the extremal case c = r + d +1, we only need to check with the unique maximalclique with the aforementioned moving strategy. Regarding this clique, we can imaginethe moving subsequence of the i -th round λ i as a virtual r -tuple λ i = ( λ i , . . . , λ ri ). It isnot difficult to check that for each j with 1 ≤ j ≤ r , we have λ ji = j, if 1 + r ≤ i + j ≤ d + 1 + r, void movement , otherwise . In other words, the moving sequence is the concatenation of the rows of the ( d + r ) × r diagram in Figure 5, starting from the top while void movements are left blank. As aquick example, consider the case when ( r, c, d ) = (3 , , Y , , : r λ : r − rr − r r − r r − λ r + d − : 1 2 λ r + d : 1 d + r o w s Figure 5. Moving sequence in the case of c = r + d + 1and consequently the final variable has to be Y , , . The above moving strategy leads tothe maximal clique (cid:110) Y , , −→ Y , , −→ Y , , −→ Y , , −→ Y , , −→ Y , , −→ Y , , (cid:111) . Since we have r + d rounds of movements and only the end movement at each roundfails to contribute any corner generator, the corresponding colon ideal contains exactly r ( d + 1) − ( r + d ) = rd − d corner generators. In other words,pd((in( P )) ∨ ) = d ( r − , agreeing with the formula in (39).4.5.2. Step (II) . In the extremal case when c = 2 r + d − 1, we apply the revlex typeordering of variables. Notice that N − c + ( r − d − r (2 + d ) − . Furthermore, when r = 2, c = 2 r + d − r + d + 1, which has been covered by Step(I). Thus, we will assume that r ≥ F , the number of corners is bound from the above by( N − − (cid:100) ( N − /r (cid:101) = ( r − d ) − , (40)which is precisely the right-side of (39) in this case. This upper bound can be achieved bythe following special maximal clique F = { Y γ > · · · > Y γ N } with the moving sequenceas a concatenation in order of • r, r − , . . . , , • d sequences of the form ( r, r − , . . . , , • r − , r − , . . . , , γ = (1 , d, d ) , . . . , ( r − d ))and γ N = (2 + d, d ) , . . . , ( r − d ) , r (2 + d ) − 1) (41)by (22), (24) and (25). As a quick example, when ( r, c, d ) = (3 , , F in mind will be (cid:110) Y , , −→ Y , , −→ Y , , −→ Y , , −→ Y , , −→ Y , , −→ Y , , −→ Y , , (cid:111) . egarding this clique, we claim that the tail generator of F will be canceled out by itscorners. Hence the minimal monomial generating set G F contains only corners. To seethis, we notice that the last movement is 1 < r . Therefore δ = N in (28). Now, by (30)and (41), the special k in (29) is at least 2. Notice that δ = N − 1, which means thatthe tail generator contains the corner generator Y γ N − . As a consequence of this claim,the minimal generating set G ( I F ) consists of only corner generators. Whence,Ess( F ) = { Y γ , Y γ r , Y γ r , . . . , Y γ ( d +1) r , Y γ N } . (42)It remains to show that the integer in (40) gives the desired projective dimension, i.e.,to verify the conditions (C1)-(C3) in Lemma 4.12. The (C1) part is automatic.As for the condition (C2), suppose that F = { Y β > · · · > Y β N } is a maximal cliquewith Ess( F ) = Ess( F ). We have seen in Remark 4.13 that Y β = Y γ . Consequently, Y β N = Y γ N ∈ Ess( F ) = Ess( F ). This implies that G ( I F ) consists of only corner gen-erators. Whence, the moving subsequences connecting each adjacent pair in Ess( F )regarding F are all strictly decreasing. Notice that each such a strictly decreasing mov-ing subsequence is completely determined by its two terminal variables in F . SinceEss( F ) = Ess( F ), the moving sequence of F agrees with that of F . This is equivalentto saying that F = F , verifying the condition (C2).It remains to verify (C3). Indeed, since other subcases are similar as in Section 4.4,we only need to consider the case when the maximal clique F (cid:48) = { Y α > · · · > Y α N } satisfies α = γ (consequently α N = γ N ) and Ess( F (cid:48) ) = Ess( F ) \ { Y α N } . Now, the final r − F (cid:48) is just a rearrangement of ( r − , r − , . . . , , r .Furthermore, since γ = 2(2 + d ), it follows from (31) that the special k in (29) for F (cid:48) isat most 2. Notice that δ ≥ N − F (cid:48) . In turn, the degree of the tail generator for F (cid:48) is at most N − ( N − 1) + 1 = 2, and it equals 2 exactly when δ = N − F (cid:48) , orequivalently, the last movement is 1.(i) When the last movement is 1, the tail generator is quadratic and all the remainingmovements (a rearrangement of ( r − , r − , . . . , , F (cid:48) ). Whence, the corresponding moving subsequence is strictlydecreasing and has to be exactly ( r − , r − , . . . , , λ F (cid:48) = λ F ,and in turn F (cid:48) = F , a contradiction.(ii) When the last movement is not 1, the tail generator is linear. However, if the lastmovement of position 1 in the moving sequence is the movement from Y α j to Y α j +1 with N − r + 1 ≤ j < N − 1, it is the final movement of some maximally strictlydecreasing subsequence. Whence, Y α j +1 ∈ Ess( F (cid:48) ) \ Ess( F ), another contradiction.And this completes our argument for the case when c = 2 r + d − Step (III) . In this subsection, we focus on the case when r + d + 1 ≤ c < r + d − (cid:96) revlexr,c,d + d ≤ (cid:96) revlexr +1 ,c +1 ,d . Whence, r ≥ 3. Obviously, we will apply the revlex type ordering of variables in the following. Let F be a maximal clique in the case( r, c, d ) such that (cid:96) F = (cid:96) revlexr,c,d . Let λ F be the corresponding moving sequence and break itinto maximally strictly decreasing subsequences λ , λ , . . . , λ s . Since the length of eachsubsequence is at most r , it is clear that s ≥ ¢ N − r • ≥ ¢ r + d + 1 + ( r − d − r • = d + 1for N = N ( r, c, d ) = c + ( r − d . Suppose that F = { Y β > · · · > Y β N } . We willconstruct a related maximal clique ‹ F = (cid:110) ‹ Y α > · · · > ‹ Y α (cid:101) N (cid:111) in the case ( r + 1 , c + 1 , d )as follows. As a reminder, the ring for ‹ F will be K [ (cid:102) Y ] = K [ ‹ Y α | α ∈ Λ r +1 ,d ( (cid:102) N )] where N = N ( r + 1 , c + 1 , d ) = c + 1 + ( r + 1 − d = N + d + 1. The ( r + 1)-tuple α will beobtained by appending β with N = c + ( r − d . The facet ‹ F is then determined by themoving sequence ( › λ , › λ , . . . , › λ s ), where • › λ = r + 1 is a sequence containing just one element, • (cid:102) λ i is obtained by prepending λ i by r + 1 for 1 ≤ i ≤ d , and • (cid:102) λ i = λ i for d + 1 ≤ i ≤ s .As a quick example, consider the case ( r, c, d ) = (3 , , F = { Y , , > Y , , > Y , , > Y , , > Y , , > Y , , > Y , , } with the moving sequence λ F = ( λ , λ , λ , λ ) = (3 , , , , , 1) is a maximal clique with (cid:96) F = (cid:96) revlex , , . The newly constructed maximal clique will be ‹ F = ¶ ‹ Y , , , > ‹ Y , , , > ‹ Y , , , > ‹ Y , , , > ‹ Y , , , > ‹ Y , , , > ‹ Y , , , > ‹ Y , , , > ‹ Y , , , © with the moving sequence λ (cid:101) F = ( › λ , › λ , › λ , › λ , › λ ) = (4 , , , , , , , › λ , › λ , . . . , › λ s ) is allowable. Equivalently, we get a legal maximal clique in thecase of ( r + 1 , c + 1 , d ).(ii) Notice that the final subsequences coincide: (cid:102) λ i = λ i for d + 1 ≤ i ≤ s . It followsthat α ≤ r (cid:101) N − i = β N − i and α r +1 (cid:101) N − i = (cid:102) N for i ≤ (cid:80) sj = d | λ j | . Here, | λ j | is the length ofthe sequence λ j .(iii) For the positions in the moving sequence of F that induce corner generators, thecorresponding positions in the moving sequence of ‹ F still induce corner generators.Meanwhile, the initial movement r +1 of (cid:102) λ i for 1 ≤ i ≤ d will contribute additionalcorner generators. And then, we have the complete list of corner generators for ‹ F .(iv) Furthermore, using the formula established in Step (II) as the base argument, byinduction on applying the expected inequality (cid:96) revlexr,c,d + d ≤ (cid:96) revlexr +1 ,c +1 ,d , we will have (cid:96) F = (cid:96) revlexr,c,d ≥ ( r − (2 r + d − − c ) − d ) − r + d − − c ) d = dr − r − d + 2 c − . (43)Now, let τ F and τ (cid:101) F be the tail generators for the maximal clique F and the newlyconstructed ‹ F respectively. If τ F / ∈ G F , then no matter whether τ (cid:101) F ∈ G (cid:101) F , we havealready (cid:96) F + d ≤ (cid:96) (cid:101) F by the above item (iii). Thus, we will assume that τ F ∈ G F in thefollowing and intend to prove that τ (cid:101) F ∈ G (cid:101) F . For that purpose, we first claim thatdeg( τ F ) ≤ − s (cid:88) j = d | λ j | . (44)Otherwise, all corner generators of F will be generated from the first d − r , and consequentlycontribute at most r − (cid:96) F ≤ ( d − r − 1) + 1 . But this contradicts the inequality in (43), since c ≥ r + d + 1 and r ≥ uppose in addition that the tail generator τ F is given by Y H with H = ¶ Y β δk > · · · > Y β N © at the end of Observation 4.5 for some k ≤ r − 1. Recall that δ k was defined in (28). Wecan similarly introduce (cid:101) δ k for ‹ F with respect to this k as (cid:101) δ k := min (cid:110) i (cid:12)(cid:12)(cid:12) α { k,...,r } i = α { k,...,r } (cid:101) N (cid:111) . It follows from the above item (ii) and the inequality (44) that N − δ k ≥ (cid:102) N − (cid:101) δ k . Noticethat α k (cid:101) δ k − = α k (cid:101) δ k or α k (cid:101) δ k − β kδ k − = β kδ k or β kδ k − α k (cid:101) δ k = β kN − ( (cid:101) N − (cid:101) δ k ) ≥ β kδ k by the above item (ii) and the inequalities (20) and (44). Since 1 + ( k − d + 1) < d β kδ k − by the inequality (30), we will also have 1 + ( k − d + 1) < d α k (cid:101) δ k − , unless α k (cid:101) δ k − + 1 = α k (cid:101) δ k = β kδ k − = β kδ k . But by the redundancy − N − δ k ≥ (cid:102) N − (cid:101) δ k and the above item (ii), we will have instead α k (cid:101) δ k − = β kN − (cid:101) N + (cid:101) δ k − ≥ β kδ k − , a contradiction.Now as 1 + ( k − d + 1) < d α k (cid:101) δ k − , by the description of the tail generators inObservation 4.5, τ (cid:101) F = ‹ Y (cid:101) H for some (cid:102) H = ß ‹ Y α (cid:101) δk (cid:48) , . . . , ‹ Y α (cid:101) N ™ where k (cid:48) ≤ k . Consequently,deg( τ F ) ≥ deg( τ (cid:101) F ). As τ F belongs to the minimal generating set G F of the colon ideal of F , τ F won’t be canceled out by the corner generators of F . By our previous description ofthe corner generators for ‹ F in item (iii), τ (cid:101) F won’t be canceled out by the corner generatorsof ‹ F . Whence, τ (cid:101) F ∈ G (cid:101) F . Since in this case, both F and ‹ F have tail generators, we haveagain shown that (cid:96) F + d ≤ (cid:96) F (cid:48) in view of the previous item (iii).Consequently, (cid:96) revlexr,c,d + d ≤ (cid:96) revlexr +1 ,c +1 ,d , as expected.4.5.4. Step (IV) . In this final step, we will construct a maximal clique F giving thedesired upper bound (cid:96) revlexr,c,d , and show as in the previous section that the projectivedimension in mind is exactly this bound. As the extremal cases when c = r + d + 1 and c = 2 r + d − r + d + 1 < c < r + d − r ≥ ε = 2 r + d − − c , c = c − ε and r = r − ε . Notice that c = 2 r + d − 1. And as 1 ≤ ε ≤ r − r ≥ 3. We start with the special “maximal” movingsequence given in Step (II). Then we can apply the argument in Step (III) successively tobuild a “maximal” moving sequence as the concatenation of the rows of the ( d + 2 + ε ) × r diagram in Figure 6, starting from the top.Let F be the corresponding maximal clique. Since the special starting maximal cliqueconsidered in Step (II) has no tail generator in the corresponding minimal generating set,so does F here. Otherwise, we would have strict inequality in some intermediate processwhen applying the constructions in Step (III). Thus, (cid:96) F > (cid:96) revlexr ,c ,d + εd = (cid:96) revlexr,c,d , a contradiction.Furthermore, by the descriptions of the corner generators in the Steps (II) and (III), itis clear that now in every maximally decreasing moving subsequence of F , only the finalmovement fail to contribute a corner generator. : r λ : r r − r − r + 2 r r + 2 r + 1 r − r + 1 r r − r r − r + 2 r + 1 r r − λ d +1+ ε : r r − λ d +2+ ε : r − d + r o w s d r o w s Figure 6. Moving sequence in the case of r + d + 1 < c < r + d − F belongto Ess( F ). Thus, if the maximal clique F (cid:48) satisfies Ess( F (cid:48) ) = Ess( F ), then F (cid:48) has thesame starting variable, final variable and same corner generators as F . Consequently, F (cid:48) has the same moving sequence as F , making F (cid:48) = F . As for the condition (C3), again,we will only need to consider the case when F (cid:48) has the same starting variable as F , byRemark 4.13.(i) Suppose that Ess( F (cid:48) ) = Ess( F ) \ { Y β N } in (C3). Now, the final r − F (cid:48) is just a rearrangement of ( r − , r − , . . . , , f in (C3) has to be the position after some k -th round of movements λ k of F for some k < d +2+ ε . Let f (cid:48) and f (cid:48)(cid:48) be the positions in Ess( F ) immediatelybefore and after f respectively. Then f (cid:48) , f (cid:48)(cid:48) ∈ Ess( F (cid:48) ) as well. Consequently, allthe positions between them in F (cid:48) will contribute corner generators for F (cid:48) , andthe corresponding movements will form a strictly decreasing subsequence λ (cid:48) of F (cid:48) , which is a rearrangement of the concatenation λ k , λ k +1 . Let m be the firstmovement in λ k +1 of F . It is clear from Figure 6 that m will appear twice in λ (cid:48) ,since r ≥ 3. This is a contradiction.And this completes our construction for Step (IV), which in turn finished the proof ofProposition 4.16.4.6. The c ≤ r + d case. Here, we consider the regularity in the degenerated case when r ≤ c ≤ r + d . We start by the extremal case when c = r + d . Lemma 4.18. Suppose that c = r + d . Then the projective dimension is given by pd((in( P )) ∨ ) = ( r − d − . Proof. It is time to go over the key parts stated in the first three subsections earlier.Recall that the dimension of F ( I ) is then rc − r + 1 = rd + 1 instead of N = c + ( r − d by Proposition 3.10. Now, the minimal monomial generating set of in( P ) ∨ is still givenby (18). However, all the maximal cliques take the form F = ¶ Y β > · · · > Y β rd +1 © , here β j = j + ( j − d for j = 1 , , . . . , r ,and β jrd +1 = N − ( r − j )( d + 1) for j = 1 , , . . . , r .Here, the index β corresponds to the main diagonal of the leftmost maximal minor of H r,c,d , and β rd +1 corresponds to the rightmost one. Surely we don’t have (22). As for(20) and (21), we only need to change the corresponding N into rd + 1.We will apply the lex type ordering to the minimal generating set G ((in( P )) ∨ ). Thedescription will be completely the same as in Section 4.2. Since all maximal cliques havecommon starting and ending variables, there is no tail generator in any of the colonideals. The description of the corners are identical to that stated in Observation 4.6. Inparticular, (in( P )) ∨ has linear quotients, and its projective dimension is achieved exactlyby the maximal cardinality of the corners sets. Undoubtedly, this maximal length isachieved by applying the strategy in Remark 4.17.Due to the explicit description of the starting variable Y β , the moving sequence is thenthe concatenation of the rows of the ( d + r − × r diagram in Figure 7, starting fromthe top. It is not difficult to see that it contains rd − ( r + d − 1) = ( r − d − 1) corners. λ : r λ : r − rr − r r − r r − λ r + d − : 1 2 λ r + d − : 1 d r o w s Figure 7. Moving sequence in the case of c = r + d In other words, the expected projective dimension is given by ( r − d − (cid:3) Now, we consider the general case. Proposition 4.19. Suppose that r ≤ c ≤ r + d . Then the projective dimension is givenby pd((in( P )) ∨ ) = ( r − c − r − , if r < c, , if r = c. Proof. When r = c , I r ( H r,c,d ) is principal of degree r . And when c = r + 1, one can checkthat all pairs of the elements in Λ r,d ( N ) are comparable. In both cases, the defining ideal P = 0 and the regularity of the fiber cone is 0.When 1 < r < r + 1 < c ≤ r + d , by the reduction before [16, Theorem 2.3], wecan reduce the ideal I r ( H r,c,d ) ⊂ R = K [ x ] to some I r ( H r,c,d (cid:48) ) ⊂ K [ x (cid:48) ] with d (cid:48) = c − r .Here, the collection of variables x (cid:48) is a subset of the original collection of variables x ,and I r ( H r,c,d (cid:48) ) K [ x ] = I r ( H r,c,d ). Whence, c = r + d (cid:48) and by Lemma 4.18, the projectivedimension is pd((in( P (cid:48) )) ∨ ) = ( r − d (cid:48) − , here P (cid:48) is the defining ideal of F ( I r ( H r,c,d (cid:48) )) as P for F ( I r ( H r,c,d )). Therefore, theoriginal projective dimension ispd((in( P )) ∨ ) = ( r − c − r − . when 1 < r < c ≤ c + d . (cid:3) Therefore, we have completed the proof for our Theorem 4.1. Remark 4.20. Knowing the extremal Betti numbers of in( P ) amounts to knowing theextremal Betti numbers of (in( P )) ∨ , by [32, Theorem 5.61]. Since (in( P )) ∨ is known tohave linear quotients, the Cohen–Macaulay type of the fiber cone F (in( I )) is simply thetop Betti number of (in( P )) ∨ . And our approach in this section then paves a road towardshandling it. In particular, one can try to characterize when F (in( I )) is Gorenstein. Butthis has already been done neatly in [16, Theorem 3.7], which shows that F (in( I )) isGorenstein if and only if c ∈ { r, r + 1 , r + d, r + d + 1 , r + d } when r ≥ d ≥ 1. The paper [16] takes advantage of the Ehrhart ring theory, whichis a standard combinatorial tool for handling this type of problems. As a quick corollary,since F ( I ) and F (in( I )) have the same Cohen–Macaulay type, the original fiber cone F ( I ) is Gorenstein if and only if the number of columns c satisfies the same requirement,as mentioned earlier in Proposition 3.10.Another thing that worth mentioning is that when c = r + d , r + d + 1 or 2 r + d , thepaper [16] computed some geometric invariant δ of the associated integral convex poly-tope. Since the corresponding fiber cone F (in( I )) is Gorenstein, this integer δ is simply − a ( F (in( I ))) by [34, Proposition 2.2], where a ( F (in( I ))) is the a -invariant introducedby Goto and Watanabe in [22, Definition 3.1.4]. Whence, we can derive the correspondingregularity for free since a ( A ) = reg( A ) − dim( A ) (45)for any standard graded Cohen–Macaulay algebra A over K , in view of the equivalentdefinition of regularity in [35, Definitions 1 and 3]. It is not surprising that the outcomeagrees with our formula in Theorem 4.1 for these three special cases.We end this section with a quick application. It is also due to the following fact. Lemma 4.21 ([11, Proposition 6.6] or [21, Proposition 1.2]) . Let I ⊂ R = K [ x , . . . , x N ] be a homogeneous ideal that is generated in one degree, say d . Assume that the fiber cone F ( I ) is Cohen–Macaulay. Then each minimal reduction of I is generated by dim( F ( I )) homogeneous polynomials of degree d , and I has the reduction number r( I ) = reg( F ( I )) . Corollary 4.22. The reduction numbers of the ideal I r ( H r,c,d ) and its initial ideal in( I r ( H r,c,d )) are given by r( I r ( H r,c,d )) = r(in( I r ( H r,c,d ))) = N − − (cid:98) ( N − /r (cid:99) , if r + d ≤ c,dr − r − d + 2 c − , if r + d < c < r + d, ( r − c − r − , if r < c ≤ r + d, , if r = c, here N = c + ( r − d . And the a -invariants of F ( I r ( H r,c,d )) and F (in( I r ( H r,c,d ))) aregiven by a ( F ( I r ( H r,c,d ))) = a ( F (in I r ( H r,c,d ))) = − − (cid:98) ( N − /r (cid:99) , if r + d ≤ c,c − r − d − , if r + d < c < r + d, − c, if r < c ≤ r + d, − , if r = c. Proof. By Theorem 3.7, Proposition 3.10 (a) and [20, Corollary 3.4], the initial algebraof the fiber cone F ( I r ( H r,c,d )) is the fiber cone F (in( I r ( H r,c,d ))), and these two algebrasare both Cohen–Macaulay. Now, it suffices to apply [8, Corollary 2.5], Proposition 3.10(c), Theorem 4.1, Lemma 4.21 and Equation (45). (cid:3) Remark 4.23. The initial ideal in( I r ( H r,c,d )) is also considered as the ( d + 1)-spreadVeronese ideal of degree r in [20]. This concept was later generalized to the class of c -bounded t -spread Veronese ideals I c, ( n,d,t ) and the class of Veronese ideals of boundedsupport I ( n,d,t ) ,k in [17]. The regularity of the particular fiber cone K [ I ( n,d, ,k ] was com-puted in [17, Proposition 5.6], which has a similar flavor as that in our Proposition 4.15.It is then natural to ask for the regularity of the fiber cone of other ideals considered in[17]. 5. Regularity of fiber cones of rational normal scrolls As an application, we want to compute the regularity of the fiber cone of a generalrational normal scroll S n ,...,n d ⊆ P c + d − . For that purpose, we need the following obser-vation. Lemma 5.1 ([4, Proposition 7.43]) . Let M be a graded Cohen–Macaulay module overthe polynomial ring K [ x , . . . , x n ] , and P M ( t ) the numerator Laurent polynomial of theHilbert series of M . Then reg( M ) = deg( P M ( t )) . Theorem 5.2. Let I n ,...,n d be the defining ideal of the rational normal scroll S n ,...,n d ⊆ P c + d − for c = n + · · · + n d . (i) The Castelnuovo–Mumford regularity of the fiber cone is given by reg( F ( I n ,...,n d )) = (cid:100) ( c + d − / (cid:101) , if d ≤ c,c − , if < c < d. (ii) The reduction number of I n ,...,n d is given by r( I n ,...,n d ) = (cid:100) ( c + d − / (cid:101) , if d ≤ c,c − , if < c < d. (iii) The a -invariant of the fiber cone is given by a ( F ( I n ,...,n d )) = (cid:100) ( c + d − / (cid:101) − c − d, if d ≤ c, − c, if < c < d. Proof. Notice that the fiber cone F ( I n ,...,n d ) is Cohen–Macaulay by Theorem 2.1. Thus,the item (ii) follows directly from the item (i) and Lemma 4.21. And similarly, the item(iii) follows from the item (i), Equation (45) and the factdim( F ( I n ,...,n d )) = c + d, if 4 + d ≤ c ,2 c − , if 2 < c < d , y [38, Proposition 3.1, Corollary 3.10].As for the item (i), just as observed in the proof of [38, Theorem 3.13], the Hilbertfunction of the fiber cone of the rational normal scroll S n ,...,n d depends only on c and d by [3, Theorem 3.7]. Whence, the fiber cones of the ideal I n ,...,n d above and theideal I ( H ,c,d ) in Section 3 have the same Laurent polynomial. Now, by the Cohen–Macaulayness of F ( I n ,...,n d ) established in Theorem 2.1, these fiber cones have the sameCastelnuovo–Mumford regularity by Lemma 5.1. Now, it remains to apply Theorem 4.1. (cid:3) Remark 5.3. It is natural to ask for the Castelnuovo–Mumford regularity of the fibercone of the secant variety of the general rational normal scroll S n ,...,n d . However, to ourbest knowledge, little is known about the defining ideal of such a fiber cone. Therefore,our current method does not apply, unfortunately. Acknowledgment. The authors are grateful to the software system Macaulay2 [23], forserving as an excellent source of inspiration. The second author is partially supported bythe “Anhui Initiative in Quantum Information Technologies” (No. AHY150200) and the“Fundamental Research Funds for the Central Universities”. References [1] W. Bruns and A. 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