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troemer@uos.de Institute of Mathematics, In the present paper, we study algebraic properties of edge ideals associated with Research Unit Data Science, plane curve arrangements via their Levi graphs. Using combinatorial properties of such Osnabrück University, Albrechtstr. 28A, 49076 Levi graphs, we are able to describe those monomial algebras being Cohen–Macaulay, Osnabrück, Germany Buchsbaum, and sequentially Cohen–Macaulay. We also consider the projective Full list of author information is dimension and the Castelnuovo-Mumford regularity for these edge ideals. We provide available at the end of the article To Jürgen Herzog on his 80th eﬀective lower and upper bounds on them. As a by-product of our study, we connect, birthday in general, various Buchsbaum properties of squarefree modules. Keywords: Edge ideals, Curve arrangements, Sequentially Cohen–Macaulay rings, Buchsbaum rings, Projective dimension, Regularity, Square free modules Mathematics Subject Classiﬁcation: 05E40, 13F55 (Primary), 05C25, 13H10, 14N20 (Secondary) 1 Introduction In the present paper, we study edge ideals associated with plane curve arrangements in the complex projective plane. Such ideals can be naturally deﬁned via the notion of a Levi graph that encodes properties of the intersection poset of an arrangement. The goal is to understand how graph theoretic invariants are encoded algebraically in these ideals. Knowing the linkage between Levi graphs and the geometry of arrangements, we are able to understand many algebraic properties of associated monomial algebras via curve arrangements and their combinatorics. This approach is the main novelty of our present work. Our choice of Levi graphs as a potential source for the poset structure for the associated monomial algebras is not accidental, and it is motivated by numerous applications. In the context of line arrangements, Levi graphs play a leading role in many subjects of current research. These graphs, which were introduced by Coxeter [4], are the bipartite graphs that decode the intersection poset structure of a given arrangement of lines L ⊂ P .More precisely, for a collection of points and lines in a projective plane, we construct a graph with one vertex per point, one vertex per line, and an edge for every incidence between a point and a line. It turns out that a Levi graph carries the same combinatorial information as the inter- section lattice, which is the reason why this object attracts a lot of attention. One of the © The Author(s) 2022, corrected publication 2022. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 0123456789().,–: volV 30 Page 2 of 17 Pokora, Römer Res Math Sci (2022) 9:30 most fundamental problems that involves line arrangements in the plane is the celebrated Terao’s conjecture on freeness of hyperplane arrangements, i.e., the freeness of the mod- ule of polynomial derivatives associated with an arrangement L is determined by its Levi graph (or equivalently, the intersection lattice). In the case of arbitrary arrangements of curves in the plane, the intersection structure, in general, does not form a lattice, but only the intersection poset. However, in some cases we can still deﬁne honest variant of the Levi graph that will decode the intersection structure, and we are going to study it in the context of d-arrangements of plane curves, a special class of smooth plane curves that naturally generalizes line arrangements in the plane. Edge ideals can be studied from the viewpoint of Stanley–Reisner rings, where the poset structure of the facets associated with the abstract simplicial complex is the crucial infor- mation that tells us about many algebraic properties. There are many interesting articles andbooksdevotedtopropertiesofedgeidealslike[26,27]. Here we are going to study edge ideals deﬁned via Levi graphs of arrangements and we want to understand algebraic prop- erties of those ideals by looking at the combinatorics of Levi graphs. From this perspective, we will be able to provide complete classiﬁcation results which is the key advantage of our approach. More precisely, the main purpose of the present manuscript is to introduce a certain class of edge ideals determined by the intersection posets of d-arrangements via the notion of their Levi graph. This will allow us to reinterpret algebraic properties of monomial algebras in the language of the geometry of arrangements. For instance, we will be able to say which intersection posets of d-arrangements of curves will provide to us examples of sequentially Cohen–Macaulay algebras (see Theorem 6.5). These results might be used in diﬀerent branches of algebraic geometry, where the presence of edge ideals and the associated bipartite graphs plays an prominent role. In the application context, it is worth mentioning that edge ideals arise naturally in the context of rigid toric varieties determined by bipartite graphs [18]. Except results devoted to edge ideals and arrangements of curves, we study general properties of squarefree modules and their homological properties. In Sect. 4, a special class of dual squarefree modules is deﬁned and prove that for squarefree S = k[x , ..., x ]-modules being Buchsbaum is equivalent 1 n to be k-Buchsbaum for every k 1(seeTheorem 4.12). This general result allows us to conclude that for edge ideals I(G) associated with bipartite graphs G the condition that the quotient algebra S/I(G) is Cohen–Macaulay is equivalent to be both Buchsbaum and k-Buchsbaum for every k 1. At the end of the paper, we study homological properties of the quotient algebras S/I(G), where I(G) is the edge ideal associated with the Levi graph of a d-arrangement, and we provide some eﬀective bounds on the projective dimension and the Castelnuovo-Mumford regularity for powers of edge ideals. Based on examples, several open problems related to the geometry of d-arrangements and homological prop- erties of the associated edge ideals are formulated. For some examples, that allow us to made our predictions, we performed symbolic computations supported by scripts written in Singular [8]. 2 Arrangements of curves In this section, we are going to deﬁne our main geometrical object of studies, namely d-arrangements of plane curves. Such arrangements can be considered as a natural gen- eralization of line arrangements, especially from the viewpoint of their singularities. Let Pokora, Römer Res Math Sci (2022) 9:30 Page 3 of 17 30 us emphasize in this place that for us singular points are just the intersection points of curves, i.e., points in the plane where at least two curves from the arrangement meet. Deﬁnition 2.1 Let C ={C , ..., C }⊂ P be an arrangement of k 3 curves in the plane. 1 k We say that C is a d-arrangement if • all curves C are smooth of the same degree d 1; • the singular locus Sing(C) consists of only ordinary intersection points – they look locally like intersections of lines. In particular, 1-arrangements are just line arrangements. However, the class of 2- arrangement is strictly smaller than the class of all conic arrangements in the plane – usually the intersection points of conic arrangements are not ordinary.If C is a d- arrangement, then we have the following combinatorial count: k m d = , (1) 2 2 p∈Sing(C) where m denotes the multiplicity, i.e., the number of curves from C passing through p ∈ Sing(C). Moreover, for every C ∈ C one has d (k − 1) = (m − 1). (2) p∈Sing(C)∩C Using Hirzebruch’s convention, for a given d-arrangement C we denote by t = t (C) r r the number of r-fold points, i.e., points in the plane where exactly r curves from the arrangement meet. Additionally, for s being the number of intersection points in Sing(C) we also have s = t . r2 One of the most important information that each d-arrangement carries is the intersec- tion poset which decodes the incidences between curves and intersection points. Since all the intersection points of d-arrangements are ordinary, they behave analytically like line arrangements around the intersection points. This observation motivates us to introduce the notion of the Levi graph for d-arrangements. Deﬁnition 2.2 Let C be a d-arrangement. Then the Levi graph G = (V, E) is a bipartite graph with V := V ∪ V ={x , ..., x ,y , ..., y }, where each vertex y corresponds to a 1 2 1 s 1 k i curve C , each vertex x corresponds to an intersection point p ∈ Sing(C)and x is joined i j j j with y by an edge in E if and only if p is incident with C . i j i The importance of the Levi graphs for d-arrangements can be observed in the context of a naive generalization of Terao’s conjecture on the freeness of d-arrangements. For the completeness of the exposition, let us present this conjecture. Conjecture 2.3 (Generalized Terao conjecture) Let C , C ⊂ P be two d-arrangements 1 2 for a ﬁxed d 1 and denote by G ,G the associated Levi graphs. Assume that G and G 1 2 1 2 are isomorphic, is it true that the freenees of C implies that C is also free? 1 2 30 Page 4 of 17 Pokora, Römer Res Math Sci (2022) 9:30 Our expectation is that this generalization of Terao’s conjecture should be false for d 2, mostly due to the fact that there are examples of conic-line arrangements in the plane with ordinary singularities providing counterexamples to a generalized Terao’s conjecture (see [21] for details). 3 Edge ideals associated with d-arrangements Let G be a simple graph with vertices x , ..., x and let S = k[x , ..., x ] be a polynomial 1 n 1 n ring over a ﬁxed ﬁeld k. We will use the convention that x will denote both a vertex of G and also a variable of S. Deﬁnition 3.1 The edge ideal I(G) associated with the graph G is the ideal of S generated by the set of all squarefree monomials x x such that x is adjacent to x .The ring S/I(G) i j i j is called the edge ring of G. In the setting of Levi graphs associated with d-arrangements, we will work in S = k[x , ..., x ,y , ..., y ] in order to emphasize the incidence structure of the Levi graph G 1 s 1 k of C,namely x ’s correspond to the intersection points, y ’s correspond to curves in a i j d-arrangement, and we have an edge joining x and y if and only if the intersection point i j P lies on the curve C in C. i j Deﬁnition 3.2 Given a poset P with the vertex set X ={x , ..., x },its order complex, 1 n denoted by , is the simplicial complex on X whose faces are the chains (linearly ordered sets) in P. With the notation of the deﬁnition, we have that k[] = k[X]/x x : x x i j i j is the Stanley–Reisner ring of , where x x means that the elements x and x are not i j i j comparable. The simplicial complex of the graph G is deﬁned by ={A ⊆ V : A is an independent set in G}, where A is an independent set in G if none of its elements are adjacent. Observe that is precisely the simplicial complex with the Stanley–Reisner ideal I(G). Recall: Deﬁnition 3.3 Let I ⊂ S be a graded ideal. The quotient ring R/I is Cohen–Macaulay if depth(R/I) = dim(R/I). The ideal I is called Cohen–Macaulay if R/I is Cohen–Macaulay. See the book [1] as a general reference on Cohen–Macaulay ideals and modules. We say that a graph G is Cohen–Macaulay over k if R/I(G) is a Cohen–Macaulay ring. It is worth noticing also that the celebrated result of Eagon and Reiner states that an abstract simplicial complex is Cohen–Macaulay if and only if the Stanley–Reisner ideal of its Alexander dual, denoted by I , has a linear resolution. We are going to show an example of a Cohen–Macaulay graph G that is associated with a point-line conﬁguration in the plane. Warning: The class of point-line conﬁgurations in the plane is strictly larger that the class of line arrangements. For us, every line arrangement is a point-line conﬁguration, but the revers statement is obviously false. For instance, for a point-line conﬁguration, a point in the conﬁguration need not to be an intersection point! Pokora, Römer Res Math Sci (2022) 9:30 Page 5 of 17 30 Example 3.4 Consider a point-line conﬁguration in the plane consisting of 3 points and 3 lines, where we have exactly one triple intersection point and two points which are sitting on two distinct lines—these two points are not intersection points! More precisely, below the Levi graph of this conﬁgurations is presented—here x ’s denote the points, and y ’s i j denote the lines: G = {x ,x ,x ,y ,y ,y }, {x ,y }, {x ,y }, {x ,y }, {x ,y }, {x ,y } . 1 2 3 1 2 3 1 1 2 1 2 2 2 3 3 3 Let S = k[x ,x ,x ,y ,y ,y ] 1 2 3 1 2 3 and consider the following edge ideal associated with G,namely I(G) =x y ,x y ,x y ,x y ,x y . 1 1 2 1 2 2 2 3 3 3 We show that the monomial algebra M := S/I(G) is Cohen–Macaulay. In order to do so, we prove that the Alexander dual I(G) has a linear resolution. First of all, one needs to ﬁnd the Alexander dual of I(G), and this can be done by computing the primary decomposition of I.Asaresult,weobtain I(G) =y y y ,y y x ,y x x ,y x x ,x x x . 1 2 3 1 3 2 3 1 2 1 2 3 1 2 3 The minimal graded free resolution of N = S/I(G) has the following form 5 5 0 → S(−5) → S (−4) → S (−3) → S → N → 0, which shows that M is a Cohen–Macaulay monomial algebra. 4 Squarefree modules and their properties The main goal of this section is to recall the notion of squarefree modules, which were introduced and studied by Yanagawa [28]. See also [19,20] for related results. We study further aspects of them needed in the following. For this we start with our general setup which follows [12, Section 1 and 2]. All modules in this sections are ﬁnitely generated n n Z -graded S-modules M =⊕ M . Such a module M is called N -graded,if M = 0 a∈Z a a for a ∈ / N .Let k be a ﬁeld and denote by S = k[x , ..., x ] the polynomial ring with the 1 n standard Z -grading and let m = (x , ..., x ) be the standard graded maximal ideal. In 1 n this section, we consider S-modules M with Z -grading as a default grading. For such a module, one deﬁnes the dual module M as N := M = Hom (M, k). Remark 4.1 The deﬁned above dual S-module N to M can be viewed from a viewpoint of the Matlis duality since we have N = Hom (M, k) Hom (M, E (k)), k S S −1 −1 where E (k) = Hom (S, k) = k[x , ..., x ] is the injective hull of k as an S-module. The S k 1 n degree b ∈ Z part of the Matlis duality is N := (M ) = Hom (M , k), b b k −b 30 Page 6 of 17 Pokora, Römer Res Math Sci (2022) 9:30 so the Matlis duality reverses the grading in the presented sense. Under the assumption that M is a ﬁnitely dimensional k-vector space for every b ∈ Z , ∨ ∨ we have the isomorphism (M ) M. Let M be the category of Z -graded S-modules and whose morphisms are homogeneous of degree 0. Observe that the multiplication by x gives the following homomorphism of k-vector spaces x : M −→ M i b b+ε for any M = M ∈ M, where ε is the i-th vector of the canonical basis of Z . n i b∈Z Using the Matlis duality, we can deﬁne the multiplication map on the duals, namely x : N → N , i b b+ε which has the following explicit form (x · φ)(m) = φ(x · m). i i Finally, let us denote the canonical module ω in the Z -grading situation, i.e., ω = S S S(−1, ..., −1) = S(−1). The graded local duality theorem [12, Theorem 2.2.2] states: Theorem 4.2 For any ﬁnitely generated Z -graded S-module M, one has n−i i ∨ H (M) = Ext (M, ω ). Let us also recall the notion of k-Buchsbaum modules from [10]. Deﬁnition 4.3 Let M be a Z -graded S-module. We say that M is k-Buchsbaum if k is k i the minimal non-negative integer satisfying m · H (M) = 0 for i < dim M. Remark 4.4 (i) Being a module M with 0-Buchsbaum property is equivalent to the fact that M is Cohen–Macaulay. (ii) The Buchsbaum property implies 1-Buchsbaumness (see [23, Corollary 2.4]). (iii) In general, being 1-Buchsbaum does not imply the Buchsbaum property (see [23, Example 2.5]) for a nice geometrical counterexample. However, in some cases, like squarefree modules, the notion of 1-Buchsbaum and Buchsbaum modules coincides. We say that a Z -graded S-module M is degreewise ﬁnite if M is a ﬁnitely dimensional k-vector space for every b ∈ Z . n ∨ k Lemma 4.5 Let M be a degreewise ﬁnite Z -graded S-module and N = M .Then m M = 0 if and only if m N = 0. Proof If m M = 0, then by our deﬁnition of the multiplication map on the dual modules k ∨ ∨ we have m N = 0. The revers implication follows from the fact that (M ) = M. Corollary 4.6 Let M be a ﬁnitely generated Z -graded S-module. Then the following statements are equivalent: (i) M is k-Buchsbaum; n−i (ii) m · Ext (M, ω ) = 0 for every i < dim M. Proof This follows from Lemma 4.5 and Theorem 4.2. Now we pass to squarefree modules. Let us recall basics on them, following Yanagawa’s approach. Pokora, Römer Res Math Sci (2022) 9:30 Page 7 of 17 30 Deﬁnition 4.7 We say that a Z -graded S-module M is squarefree if the following con- ditions are satisﬁed: (i) M is ﬁnitely generated; (ii) M =⊕ M ; a∈N a (iii) the map M y → x y ∈ M is bijective for all a ∈ N and i ∈ supp(a). a i a+ε For example, a Stanley–Reisner ring k[] is a squarefree S-module. Moreover, if M and N are squarefree S-modules and f : M → N is a degree-preserving map, then both ker(f ) and coker(f ) are squarefree – see Yanagawa’s paper [28] for details. One immediately sees: n ∨ Lemma 4.8 Let M be a ﬁnitely generated Z -graded S-module and N = M . Then the following statements are equivalent: (i) M is squarefree; (ii) The map N y → x y ∈ N is bijective for all b ∈−N and i ∈ supp(b),and b−ε b N = 0 provided that b ∈− / N . Deﬁnition 4.9 (Dual squarefree module) Let M be a Z -graded S-module. We say that M is a dual squarefree module if the following conditions are satisﬁed: (i) M is Artinian; (ii) N = 0 provided that b ∈− / N ; (iii) The multiplication map N y → x y ∈ N is bijective for all b ∈−N and b−ε i b i ∈ supp(b). Corollary 4.10 If M is a squarefree S-module, then H (M)aredualsquarefreeS-modules for all i. Proof Observe that if M is a squarefree S-module, then also Ext (M, ω ) is squarefree for all i, which follows from Yanagawa’s paper [28]. The claim of the corollary follows from Lemma 4.8 and Theorem 4.2. Lemma 4.11 Let M be a Z -graded S-module. (i) If M is either squarefree or dual squarefree and dim M < ∞,thenM = 0 for any k b b ∈ Z \{0}. 1 n (ii) If M is dual squarefree, then m M = 0 if and only if M = 0 for any b ∈ Z \{0}. The ﬁrst main result of this paper connects various Buchsbaum properties of squarefree modules as follows: Theorem 4.12 Let M be a squarefree module over S = k[x , ..., x ] and d = dim M. Then 1 n the following statements are equivalent: (i) M is Buchsbaum; 1 n−i (ii) Mis 1-Buchsbaum, or equivalently m · Ext (M, ω ) = 0 for every i < d; k n−i (iii) M is k-Buchsbaum for every k 1, or equivalently m · Ext (M, ω ) = 0 for every i < dand k 1; k n−i (iv) M is k-Buchsbaum for some k 1, or equivalently m · Ext (M, ω ) = 0 for every i < dand for some k 1. 30 Page 8 of 17 Pokora, Römer Res Math Sci (2022) 9:30 Proof The dual statements follow all from Corollary 4.6 and in the following we concen- trate on the equivalence of the ﬁrst parts of each statement (i) to (iv). (i) ⇒ (ii): If M is Buchsbaum, then by [28, Corollary 2.7, (a) ⇒ (d)] we have i i H (M) = H (M) for any i = 0, ... ,d − 1 m m 1 i and thus m H (M) = 0 by Lemma 4.11 (ii). Hence, M is 1-Buchsbaum. (ii) ⇒ (i): If (ii) holds, then by Corollary 4.10 and Lemma 4.11 (ii) we have that i i H (M) = H (M) for any i = 0, ... ,d − 1. m m Then (i) follows from [28, Corollary 2.7, (d) ⇒ (a)]. (ii) ⇒ (iii), (iii) ⇒ (iv): These implications are trivially true by deﬁnitions. (iv) ⇒ (ii): Assume that M is k-Buchsbaum for some k 1. Next, we assume that for some i ∈{0, ... ,d − 1} there exists a vector b ∈−N such that H (M) = 0. Choose j ∈ supp(b). Then it follows that H (M) = 0, by Corollary 4.10. This yields m b−kε the contradiction k i m H (M) = 0. m b−kε i i Hence, H (M) = H (M) and (ii) follows from Lemma 4.11 (ii). m m 5 Buchsbaumness for Levi graphs of line arrangements In the main results of this section, we study in particular the relationship of various ring properties considered so far for edge rings of d-arrangements. Our proof is based on three steps. First of all, we are going to use a result due to Herzog and Hibi which provides a combinatorial description of edge ideals associated with bipartite graphs, and a result due to Cook II and Nagel about the equivalence of Cohen–Macaulay and Buchsbaum algebras associated with bipartite graphs. We sum up these results in the forthcoming theorem. Theorem 5.1 Let G be a bipartite graph with the partition V ={x , ..., x } and V = 1 1 n 2 {y , ..., y }. Then the following conditions for S/I(G) are equivalent: 1 n (i) S/I(G) is Cohen–Macaulay; (ii) S/I(G) is Buchsbaum for non-complete bipartite graph G; (iii) S/I(G) is k-Buchsbaum for some k 0; (iv) n = n and there exists a re-ordering of the sets of vertices V ,V such that 1 2 (a) x y ∈ Efor all i, i i (b) if x y ∈ E, then i j, i j (c) if x y and x y areinE,thenx y ∈ E; i j j i k k (v) G has a cross-free pure order. Remark 5.2 We say that the partitioning and ordering of vertices in G satisfying a)and b)in(iv) above is a pure order of G. Furthermore, we say that a pure order has a cross if, for some i = j, x y and x y are edges of G, otherwise we say the order is cross-free. i j j i Pokora, Römer Res Math Sci (2022) 9:30 Page 9 of 17 30 Proof (i) ⇔ (ii) follows from [3], (i) ⇔ (iv) is obtained from [14], and (iv) ⇔ (v) proven again in [3]. We need to show (ii) ⇔ (iii), but this follows from the fact that S/I(G)is squarefree and then we can apply Theorem 4.12 (ii) ⇔ (iii). Lemma 5.3 In the setting of the above characterization, a necessary condition that the edge ideal determined by the Levi graph G of a line arrangement L is Cohen–Macaulay is s = t = k. r2 A classical result due to de-Bruijn and Erdos ˝ [7] provides a complete classiﬁcation of such line arrangements with s = k. Theorem 5.4 (de-Bruijn–Erdos) ˝ Let L ⊂ P be an arrangement of k 3 lines in the plane such that t = 0,where k is an arbitrary ﬁeld. Then s k and the equality holds if and only if L is either • a Hirzebruch quasi-pencil consisting such that t = 1 and t = k − 1,or k−1 2 2 • a ﬁnite projective plane arrangement consisting of q + q + 1 points and q + q + 1 lines, where q = p for some prime number p ∈ Z . Based on de-Bruijn and Erdos ˝ theorem we can formulate the following. Theorem 5.5 Let k be any ﬁeld and let L ⊂ P be an arrangement of k 3 lines with s intersection points. Denote by I(G) the edge ideal determined by Levi graph associated with L.Then k[x , ..., x ,y , ..., y ]/I(G) is never Cohen–Macaulay. 1 s 1 k Proof We are going to show that both Hirzebruch quasi-pencils and ﬁnite projective plane do not satisfy Herzog-Hibi’s criterion. We start with the case when L ={ , ..., } is a quasi-pencil. Assume that V ={x , ..., x } corresponds to the set of intersection 1 1 k points of L and V ={y , ..., y } corresponds to the set of lines in L. Take any ordering 2 1 k satisfying the ﬁrst two conditions in Herzog-Hibi’s criterion. Without lost of generality, we may assume the following conditions (up to relabelling of vertices): • x corresponds to the point P of multiplicity d − 1; 1 1 • y , ..., y are the elements corresponding lines , ..., passing through the point 1 k−1 1 k−1 P and P ∈ / ; 1 1 k • y corresponds to ; k k • P is a double intersection point and x is corresponding element to P ; i i i • y ,y are the elements corresponding to lines intersecting at P . i k i Then obviously x y and x y are the edges of G,but x y does not correspond to any edge 1 i i 1 k k in the Levi graph G since P is not incident with line . This shows that the associated algebra cannot be Cohen–Macaulay. Next we consider the case of a ﬁnite projective plane F which has exactly q + q + 1 points and q + q + 1 lines. Assume that V ={x , ..., x 2 } corresponds to the set 1 1 q +q+1 of intersection points of L and V ={y , ..., y 2 } corresponds to the set of lines in 2 1 q +q+1 the arrangement. Take any ordering satisfying the ﬁrst two conditions in Herzog-Hibi’s criterion. We are going to show that the third condition of the mentioned criterion is not satisﬁed. To this end, consider the point (1 : 1 : 1) and we denote by x the corresponding element. Then take the line (q − 1)x + z = 0 and the corresponding element y , the point j 30 Page 10 of 17 Pokora, Römer Res Math Sci (2022) 9:30 (q − 1 : 0 : 1) and the corresponding element x , and we take ﬁnally the line y = 0with the corresponding element y . Observe that x y and x y are the edges of the Levi graph k i j j k G,but x y is not any edge of G since the point (1, 1, 1) is not incident with the line y = 0. i k This concludes the proof. Example 5.6 Unfortunately, we do not know how to extend the above result to the case of any arbitrary d-arrangement with d 2. One can show that if C is a d-arrangement with d 2, k 3and t = 0, then by [17, Lemma 4.3] we have s = t k, r2 but we do not have a global description of d-arrangements with s = k.Ifwerestrict our attention to 2-arrangements, we can easily construct an arrangement consisting of 6 conics and 6 intersection points of multiplicity 5 – just take 6 general points and all smooth conics determined by subsets consisting of 5 distinct points. We call such a conﬁguration a(6 , 6 )-symmetric point-conic conﬁguration. Let us consider the edge ideal associated 5 5 with the above arrangement I(G) =x y ,x y ,x y ,x y ,x y ,x y ,x y ,x y ,x y ,x y , 1 1 1 2 1 3 1 4 1 5 2 1 2 2 2 3 2 4 2 6 x y ,x y ,x y ,x y ,x y ,x y ,x y ,x y ,x y ,x y , 3 1 3 2 3 3 3 5 3 6 4 1 4 2 4 4 4 5 4 6 x y ,x y ,x y ,x y ,x y ,x y ,x y ,x y ,x y ,x y . 5 1 5 3 5 4 5 5 5 6 6 2 6 3 6 4 6 5 6 6 Using Singular one can compute ﬁrstly the minimal graded free resolution of the algebra k[x , ..., x ,y , ..., y ]/I(G). 1 6 1 6 The Betti diagram has the following form: 0123456789 10 ---------------------------------------------------------------------------- 0:1---------- 1: - 30 120 210 180 62----- 2: - - 15 120 400 720 765 500 204 48 5 ---------------------------------------------------------------------------- total: 1 30 135 330 580 782 765 500 204 48 5 Based on this, we can conclude that the regularity of the algebra is equal to 2, and the projective dimension is equal to 10. Moreover, we check directly that the algebra is not Cohen–Macaulay. Problem 5.7 Classify all d-arrangements with d 2 such that s = k. On the other hand, it is natural to ask whether there exists a d-arrangement with d 2 such that its algebra is Cohen–Macaulay. We will come back to this question in the forthcoming section – it turns out that such a d-arrangement does not exist. 6 Sequentially Cohen–Macaulay algebras and d-arrangements As we saw in the previous sections, edge ideals associated with line arrangements are neither Cohen–Macaulay nor k-Buchsbaum for some k 1. It is natural to wonder how this situation looks like if we focus on generalizations of Cohen–Macaulay rings. Here we focus on sequentially Cohen–Macaulay rings. Pokora, Römer Res Math Sci (2022) 9:30 Page 11 of 17 30 Deﬁnition 6.1 Let S = k[x , ..., x ]. A graded S-module M is called sequentially Cohen– 1 n Macaulay over k if there exists a ﬁnite ﬁltration of graded S-modules 0 = M ⊂ M ⊂ ... ⊂ M = M 0 1 r such that each M /M is Cohen–Macaulay and the Krull dimensions of the quotients i i−1 are increasing dim (M /M ) < dim(M /M ) < ... < dim(M /M ). 1 0 2 1 r r−1 Deﬁnition 6.2 Let G be a graph whose independence complex is . (i) We say that G is shellable graph if is a shellable simplicial complex. (ii) The graph G is called sequentially Cohen–Macaulay if the algebra S/I(G) is sequen- tially Cohen–Macaulay. In this section, we are going to provide a complete classiﬁcation of sequentially Cohen– Macaulay algebras associated with edge ideals of Levi graphs. In order to do so, we need the following two results which come from a paper by van Tuyl and Villarreal [24]. Lemma 6.3 Let G be a bipartite graph with bipartition {x , ..., x }, {y , ..., y }.IfG is 1 m 1 n shellable and G has no isolated vertices, then there is v ∈ V (G) with deg(v) = 1. Theorem 6.4 Let G be a bipartite graph. Then G is shellable if and only if G is sequentially Cohen–Macaulay. Now we are ready to provide our next classiﬁcation and main result of our work. Theorem 6.5 Let C be a d-arrangement of k 3 curves in P with s intersection points and let I(G) be the associated edge ideal determined by the Levi graph of C. Then the following conditions are equivalent: (i) k[x , ..., x ,y , ..., y ]/I(G) is sequentially Cohen–Macaulay; 1 s 1 (ii) C is a pencil of k lines in the plane. Proof First of all, it is clear that if L is a pencil of k lines, then the associated Levi graph is a tree with one root x corresponding to the d-fold point P and k leafs y , ..., y corresponding 1 1 k to lines passing through P. Since the graph K is shellable, then the associated monomial 1,k algebra k[x ,y , ..., y ]/I(G) is sequentially Cohen–Macaulay. 1 1 n From now on, we assume that C is a line arrangement with t = 0ora d-arrangement with d 2 and assume that k[x , ..., x ,y , ..., y ]/I(G) is sequentially Cohen–Macaulay. 1 s 1 We are going to apply Lemma 6.3, i.e., our aim is to show that in that setting the Levi graph of C has vertices of degree greater or equal to 2, so G is not shellable. We assume that V ={x , ..., x } corresponds to the intersection points of C and V ={y , ..., y } 1 1 s 2 1 k corresponds to the curves in C.If C is a line arrangement with t = 0, then each intersection point has multiplicity at least 2, so deg(x ) 2. Since t = 0, then by using (2)with d = 1, i k we see that on each line we have at least 2 intersection points from Sing(C) which means that deg(y ) 2, and it completes the proof for case of line arrangements. j 30 Page 12 of 17 Pokora, Römer Res Math Sci (2022) 9:30 Suppose that C is a d-arrangement with d 2. Then each intersection point has multiplicity at least 2 and this gives deg(x ) 2. Moreover, using the combinatorial count (2), we can observe that deg(y ) d . This completes the proof. As a corollary, we obtain the following classiﬁcation result. Corollary 6.6 Let C ⊂ P be a d-arrangement of k 3 curves with d 2 and s inter- section points. Denote by I(G) the edge ideal determined by Levi graph associated with C. Then k[x , ..., x ,y , ..., y ]/I(G) is never Cohen–Macaulay. 1 s 1 k Proof Let us recall that for algebras being Cohen–Macaulay implies being sequentially Cohen–Macaulay. Since for d-arrangements with d 2 the associated Levi graphs are never shellable, then the associated algebras k[x , ..., x ,y , ..., y ]/I(G) are not sequentially 1 s 1 Cohen–Macaulay. This completes the proof since pencil of lines are not d-arrangements with d 2. 7 Bounds on the projective dimension This section is motivated by the following result due to Dao, Huneke, and Schweig [6, Corollary 5.4]. Proposition 7.1 Let G be a graph on n vertices, and assume that m is the maximal degree of any vertex. Then pd(S/I(G)) n · 1 − . 2m Using the above result, we are going to provide a reasonable upper-bound for the projective dimension of an edge ideals associated with a d-arrangement. Proposition 7.2 Let C be a d-arrangement of k 3 curves having s intersection points such that t = 0.Let I(G) be the associated edge ideal in S = k[x , ..., x ,y , ..., y ].Then 1 s 1 k k s + k 1 pd(S/I(G)) (s + k) 1 − . 2 2d (k − 1) Proof If p ∈ Sing(C), then its multiplicity can be at most k − 1since t = 0. On the other hand, using Eq. (2), one can have at most d (k −1) intersection points on each curves. This 2 2 gives for the maximal degree of any vertex m max {k − 1,d (k − 1)}= d (k − 1). The stated lower bound follows directly from [9, Lemma 3.1], and this completes the proof. Example 7.3 (Projective dimension versus regularity) Let us consider an arrangement of 5 general lines in the plane, i.e., such an arrangement has only double points as the intersec- tions. We have altogether 10 such points. Let us denote by x the variable corresponding to the intersection points and by y the variables corresponding to lines. The edge ideal I(G)in S = k[x , ..., x ,y , ..., y ] of the Levi graph G has the following presentation 1 10 1 5 I(G) =x y ,x y ,x y ,x y ,x y ,x y ,x y ,x y ,x y ,x y ,x y ,x y ,x y ,x y , 1 1 1 2 2 2 2 3 3 2 3 4 4 2 4 5 5 4 5 5 6 1 6 4 7 3 7 4 x y ,x y ,x y ,x y ,x y ,x y . 8 5 8 3 9 1 9 3 10 1 10 5 Pokora, Römer Res Math Sci (2022) 9:30 Page 13 of 17 30 Using Singular we can compute the minimal graded free resolution of the algebra S/I(G). The Betti diagram has the following form: 0123456789 10 11 ---------------------------------------------------------------------------- 0: 1----------- 1: - 20 40 205------- 2: -- 90 360 515 330 100 10---- 3: - - - 80 470 1135 1370 897 295 40 - - 4: ----5 35 105 185 205 120 35 4 ---------------------------------------------------------------------------- total: 1 20 130 460 995 1500 1575 1092 500 160 35 4 We can easily see that the regularity of the algebra S/I(G) is equal to 4 and the projective dimension is is equal to 11. Notice that Corollary 7.2 tells us that the projective dimension of S/I(G) sits in the interval [8, 13.125]. Based on many similar examples (i.e., arrangements of k 3 general lines) we can observe that the regularity of the edge ideals is smaller than the number of lines (and this follows from our general bound from the previous section), but the projective dimension grows as fast as the number of double intersection points which makes the Betti diagram wide- rectangular. It is quite natural to believe that the lower bound for the projective dimension is related to the number of intersection points of a given arrangements. In this context, we propose: Problem 7.4 Let L be an arrangement of k 3 general lines in the plane and denote by I(G) the associated edge ideal. Is it true that pd (S/I(G)) ? 8 Castelnuovo–Mumford regularity for powers of edge ideals of d-arrangements After studying classical algebraic properties of modules, we are going back to homological properties. Now we pass towards a general regularity bound. Let us consider: Example 8.1 Consider a Hirzebruch quasi-pencil L of k lines with the associated Levi graph G. As usually, V ={x , ..., x } denotes the vertices corresponding to the intersection 1 1 k points and V ={y , ..., y } corresponds to the lines in L . The associated edge ideal in 2 1 k k S := k[x , ..., x ,y , ..., y ] can be presented as follows: k 1 k 1 k I(G) = x y , ..., x y ,x y ,x y , ..., x y ,x y ,x y , ..., x y ,x y . k 1 1 k k 1 2 1 3 1 k−1 2 k 3 k k−1 k k 1 By the previous results, we know that the algebra A = S /I(G) is neither Cohen– k k k Macaulay nor sequentially Cohen–Macaulay. However, it is still interesting to check other homological properties of A . First of all, we want to ﬁnd a reasonable bound on the Castelnuovo-Mumford regularity of A . There are some results providing the upper- bounds for regularity of edge ideals which use the combinatorial description of graphs. 30 Page 14 of 17 Pokora, Römer Res Math Sci (2022) 9:30 Há and van Tuyl in [13] showed that for any graph G the regularity of edge ideal I(G)is bounded from above by reg(I(G)) ν(G) + 1, where ν(G) is the matching number of a graph G – this is the maximum number of pairwise disjoint edges. The matching number of the Levi graph associated with L is equal to k and this is easy to see. This allows us to conclude that for every k 3 one has reg(A ) k. This example has a global manifestation that is valid for all d-arrangements. In order to present our main results, which are corollaries from what we will see in a moment, we need the following direct application of Hall’s theorem. Proposition 8.2 Let C be a d-arrangement with d 1 and k 3. Assume that t = 0 and denote by G the associated Levi graph with the bipartition {x , ..., x }, {y , ..., y }.Then 1 s 1 k G has matching of size k. Proof According to Hall’s theorem, we need to check whether for any subset S ⊂ {y , ..., y } we have |N(S)| |S|, where 1 k N(S) ={x : ∃ y ∈ S with {x ,y }∈ E}. i j i j Since G is connected and for each y we have that deg(y ) d + 1, then for every i i S ⊂{y , ..., y } with |S|= one has |N(S)| (d + 1) · 2, which completes the proof. Corollary 8.3 (A global regularity bound) Let C be a d-arrangement of k 3 curves with d 1 in P with s intersection points, and assume that t = 0.Let I(G) be the associated edge ideal determined by the Levi graph G of C.Then reg(I(G)) k + 1. Proof Observe that Proposition 8.2 provides us the maximal value of the matching num- ber ν(G) of the Levi graph G. First of all, Levi graphs G satisfy the following properties on the degrees of vertices: ∀ i ∈{1, ..., s},j ∈{1, ..., k} :deg(x ) 2and deg(y ) 2. i j Now the maximality of our choice follows from a simple incidence property – if we add another edge, then such an edge cannot be disjoint from the matching edges since each vertex of G has degree greater of equal to 2. Now we apply [13] to conclude the proof. It is natural to wonder whether our upper bound is close to the real value of the regularity. Example 8.4 Consider the case of a Hirzebruch quasi-pencil with k = 4 and observe that our global upper bound gives that the regularity of I(G) is bounded by 5. Using Singular, we can compute the minimal graded free resolution and the Betti diagram has the following shape: Pokora, Römer Res Math Sci (2022) 9:30 Page 15 of 17 30 --------------------------------------------------- 0: 1------ 1: -9 122--- 2: --9 24 1861 3: ---121- --------------------------------------------------- total: 1 9 21 27 20 7 1 This means that the regularity of the edge ideal I(G) is equal to 4. Example 8.5 If L is a pencil of k lines in the plane, then by Fröberg’s characterization [11] we have reg(I(G)) = 2 showing that the upper bound is sharp. This can be also explained in a slightly diﬀerent way since I(G) is sequentially Cohen–Macaulay and then by [25] we always have reg(I(G)) = ν(G) + 1. This observation motivates the following question. Problem 8.6 Let L be a d-arrangement with k 3 with the associated edge ideal I(G). Is it true that reg(I(G)) = 2implies that L is a pencil of lines? Now we focus on powers of edge ideals. We start with revisiting some results related to ordinary powers of edge ideals. Let us recall that for a homogeneous ideal I in a polynomial ring, reg(I ) is a linear function for m 0, it means that there exist integers a, b, m such that reg(I ) = am + b for all m m . This is a content of the result proved by Cutkosky et al. [5]. It is known that a is bounded above by the maximum of degrees of elements in a minimal set of generators of I,but b and m are in general diﬃcult to detect. If we restrict our attention to an edge ideals I(G) associated with any graph, then ν (G) + 1 reg(I(G)) co-chord(G) + 1, where ν (G) is the induced matching number of G, and co-chord(G) denotes the co- chordal cover number of G. These inequalities follow from [16,29]. Now we are going to focus on reg(I ) for some m 2. Our observation comes from Proposition 8.2 and results due to Jayanthan and Selvaraja [15]. Corollary 8.7 Let C be a d-arrangement with d 1,k 3, and such that k s. Denote by G the associated Levi graph. Then reg(I(G) ) 2q + k − 1. Proof This follows directly from [15, Theorem 4.4] and the fact that co-chord(G) k. Remark 8.8 • It is very natural to wonder whether there is a reasonable upper bound on the regularity of symbolic powers of edge ideals. Let us recall that by [22,Theorem 30 Page 16 of 17 Pokora, Römer Res Math Sci (2022) 9:30 5.9] for every bipartite graph G and m ∈ N one has (m) m I(G) = I(G) , which means that we have automatically obtained (q) reg(I(G) ) 2q + k − 1. • It is natural to wonder whether we can ﬁnd a reasonable upper-bound on the regularity of squarefree powers of edge ideals. This problem, in its whole generality, is open. We conclude this section by an observation regarding the Rees algebra of the edge ideal. i i Let us recall that for a given ideal I the Rees algebra R(I) = I t ⊂ R[t]. i=0 Corollary 8.9 Let C be a d-arrangement with k 3 and denote by I(G) the associated edge ideal with the Levi graph G. Then reg(R(I(G))) = k. Proof This follows from [2, Theorem 1.2] and Proposition 8.2. Acknowledgements The project was conducted when the ﬁrst author was visiting twice the University of Osnabrück in the framework of DAAD activities, namely the Personal Funding Programme Research Stays for University Academics and Scientists in February 2021, and in July 2021 as the part of the Institutional Programme Ostpartnerschaften. We would like to thank anonymous referees for comments that allowed to improve the paper. Funding Open Access funding enabled and organized by Projekt DEAL. Data availability This is not applicable as the results presented in this manuscript rely on no external sources of data or code. Author details 1 2 Department of Mathematics, Pedagogical University of Kraków, Podchora˙ ¸zych 2, 30-084 Kraków, Poland, Institute of Mathematics, Research Unit Data Science, Osnabrück University, Albrechtstr. 28A, 49076 Osnabrück, Germany. 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Research in the Mathematical Sciences – Springer Journals
Published: Jun 1, 2022
Keywords: Edge ideals; Curve arrangements; Sequentially Cohen–Macaulay rings; Buchsbaum rings; Projective dimension; Regularity; Square free modules; 05E40; 13F55 (Primary); 05C25; 13H10; 14N20 (Secondary)
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