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In this paper, a curve is any projective scheme of pure dimension one. It is well known that the arithmetic genus and the degree of an aCM curve D in P 3 are computed by the h-vector of D. However, for a given curve D in P 3 , the two aforementioned invariants of D do not tell us whether D is aCM or not. If D is an aCM curve on a smooth surface X in P 3 , any member of the linear system | D + l C | is also aCM for each non-negative integer l, where C is a hyperplane section of X. By a previous work, if a non-zero effective divisor D of degree d and arithmetic genus g on a smooth quintic surface X in P 3 is aCM and satisfies the condition h 0 (O X (D - C)) = 0 , then 0 ≤ d + 1 - g ≤ 4 . In this paper, we classify non-aCM effective divisors on smooth quintic surfaces in P 3 of degree d and arithmetic genus g such that 0 ≤ d + 1 - g ≤ 4 , and give several examples of them. [ABSTRACT FROM AUTHOR]

The h-vectors of homogeneous rings are one of the most important invariants that often reflect ring-theoretic properties. On the other hand, there are a few examples of edge rings of graphs whose h-vectors are explicitly computed. In this paper, we compute the h-vector of a special family of graphs, by using the technique of initial ideals and the associated simplicial complex. [ABSTRACT FROM AUTHOR]

Studying an (α, β)-metrics is a central idea in Finsler geometry, which is a generalization of Randers metric. In this paper, we have derived the Cartan connection for the Finsler space whose metric is given by h-Randers exponential change and also obtained the condition under which the Finslerian hypersurface to be hyperplane of first, second and third kind. [ABSTRACT FROM AUTHOR]

A very well-covered graph is a well-covered graph without isolated vertices such that the height of its edge ideal is half of the number of vertices. In this survey article, we gather together most of the old and new results on the edge and cover ideals of these graphs. [ABSTRACT FROM AUTHOR]

We verify the g-conjecture for interval subdivisions of Cohen–Macaulay simplicial complexes, using purely combinatorial methods. More precisely, we show that the g-vector of the interval subdivision of a Cohen–Macaulay simplicial complex is an f-vector. Murai defined the g ″ -vector, starting from the h ″ -vector introduced by Novik for Buchsbaum simplicial complexes. We prove that the g ″ -vector of the interval subdivision of a Buchsbaum simplicial complex is an f-vector of some simplicial complex. [ABSTRACT FROM AUTHOR]

V.F. Molchanov considered the Hilbert series for the space of invariant skew-symmetric tensors and dual tensors with polynomial coefficients under the action of a real reflection group, and he speculated that it had a certain product formula involving the exponents of the group. We show that Molchanov's speculation is false in general but holds for all coincidental complex reflection groups when appropriately modified using exponents and co-exponents. These are the irreducible well-generated (i.e., duality) reflection groups with exponents forming an arithmetic progression and include many real reflection groups and all non-real Shephard groups, e.g., the Shephard-Todd infinite family G(d, 1, n). We highlight consequences for the q-Narayana and q-Kirkman polynomials, giving simple product formulas for both, and give a q-analogue of the identity transforming the h-vector to the f-vector for the coincidental finite type cluster/Cambrian complexes of Fomin–Zelevinsky and Reading. We include the determination of the Hilbert series for the non-coincidental irreducible complex reflection groups as well. [ABSTRACT FROM AUTHOR]

We show that the γ -vector of the interval subdivision of a simplicial complex with a nonnegative and symmetric h-vector is nonnegative. In particular, we prove that such γ -vector is the f-vector of some balanced simplicial complex. Moreover, we show that the local γ -vector of the interval subdivision of a simplex is nonnegative; answering a question by Juhnke-Kubitzke et al. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new class of Finsler metrics which are not always (α; β)-metrics. We obtain the spray coefficients and Cartan connection of these metrics. We have also found a necessary and sufficient condition for them to be projective. Finally, under some suitable conditions, we obtain many new Douglas metrics from the given one. [ABSTRACT FROM AUTHOR]

We study the combinatorial structure of the poset H consisting of h-vectors of length s of codimension c standard determinantal schemes, defined by the maximal minors of a t × (t + c − 1) homogeneous, polynomial matrix. We show that H obtains a natural stratification, where each strata contains a maximum h-vector. Moreover, we prove that any h-vector in H is bounded from above by a h-vector of the same length and which corresponds to a codimension c level standard determinantal scheme. Furthermore, we show that the only strata in which there exists also a minimum h-vector is the one consisting of h-vectors of level standard determinantal schemes.

We have considered the β-change of Finsler metric L given by L = f(L, β) where f is any positively homogeneous function of degree one in L and β. Here β = bi(x, y)yi, in which bi are components of a covariant h-vector in Finsler space Fn with metric L. We have obtained that due to this change of Finsler metric, the imbedding class of their tangent Riemannian space is increased at the most by two. [ABSTRACT FROM AUTHOR]

For a poset P and an integer r ≥ 1 , let P r be the collection of all r -multichains in P. Corresponding to each strictly increasing map ı : [ r ] → [ 2 r ] , there is an order ⪯ ı on P r. Let Δ (G ı (P r)) be the clique complex of the graph G ı associated to P r and ı. In a recent paper [14] , it is shown that Δ (G ı (P r)) is a subdivision of P for a class of strictly increasing maps. In this paper, we show that all these subdivisions have the same f -vector. We give an explicit description of the transformation matrices from the f - and h -vectors of Δ to the f - and h -vectors of these subdivisions when P is a poset of faces of Δ. We study two important subdivisions, namely Cheeger-Müller-Schrader's subdivision and the r -colored barycentric subdivision which fall in our class of r -multichain subdivisions. [ABSTRACT FROM AUTHOR]

Given two h-vectors, h and h', we study which are the possible h-vectors for the union of two disjoint sets of points in P^2 , respectively associated to h and h' and how they can be constructed. We will give some bounds for the resulting h-vector and we will show how to construct the minimal h-vector of the union among all possible ones.

Let R = k [ T 1 , … , T f ] be a standard graded polynomial ring over the field k and Ψ be an f × g matrix of linear forms from R , where 1 ≤ g < f . Assume [ T 1 ⋯ T f ] Ψ is 0 and that grade I g ( Ψ ) is exactly one short of the maximum possible grade. We resolve R ‾ = R / I g ( Ψ ) , prove that R ‾ has a g -linear resolution, record explicit formulas for the h -vector and multiplicity of R ‾ , and prove that if f − g is even, then the ideal I g ( Ψ ) is unmixed. Furthermore, if f − g is odd, then we identify an explicit generating set for the unmixed part, I g ( Ψ ) unm , of I g ( Ψ ) , resolve R / I g ( Ψ ) unm , and record explicit formulas for the h -vector of R / I g ( Ψ ) unm . (The rings R / I g ( Ψ ) and R / I g ( Ψ ) unm automatically have the same multiplicity.) These results have applications to the study of the blow-up algebras associated to linearly presented grade three Gorenstein ideals. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to obtain the relation between imbedding class numbers of tangent Riemannian spaces to (Mn,L) and (Mn,L*) where L*(x, y) is obtained from the transformation of L(x, y) is given by L*(x, y) → L(x, y) + bi(x, y)yi [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of the spanning simplicial complex Δs(G) associated to a simple finite connected graph G. We characterize all spanning trees of the uni-cyclic graph Un,m. In particular, we give a formula for computing the Hilbert series and h-vector of the Stanley-Reisner ring k[Δs(Un,m)]. Finally, we prove that the spanning simplicial complex Δs(Un,m) is shifted and hence is shellable. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to find the necessary and sufficient conditions under which a generalized h-Kropina change of Finsler metric becomes a projective change. The condition under which a generalized h-Kropina change of Finsler metric of Douglas space leads to a Douglas space have also been found. [ABSTRACT FROM AUTHOR]

Let G be a Cohen-Macaulay very well-covered graph. We prove that the h-vector of the independence complex of G is precisely the f-vector of a flag complex. Moreover, we show that the h-vector of clique-whiskered graphs is exactly the h-vector of Cohen-Macaulay very well-covered graphs. In particular, we show that Kalai's conjecture holds if G is a very well covered Cohen-Macaulay graph. [ABSTRACT FROM AUTHOR]

The family of Buchsbaum simplicial posets generalizes the family of simplicial cell manifolds. The $h'-$vector of a simplicial complex or simplicial poset encodes the combinatorial and topological data of its face numbers and the reduced Betti numbers of its geometric realization. Novik and Swartz showed that the $h'-$vector of a Buchsbaum simplicial poset satisfies certain simple inequalities. In this paper we show that these necessary conditions are in fact sufficient to characterize the h'-vectors of Buchsbaum simplicial posets with prescribed Betti numbers.

In this note we consider the sequence whose nth term is the number of M-sequences of length n. We show that the nth term of this sequence is bounded above by the nth Fibonacci number and bounded below by the number of integer partitions of n into distinct parts. [ABSTRACT FROM AUTHOR]

For a complete intersection setof points in a projective space ℙnover an algebraically closed field, we prove that theh-vector of the graded Artinian ringR/(I𝕏 + I𝕐) is an SI-sequence provided 𝕏 is also a complete intersection subset of ℤ and 𝕐 = ℤ \ 𝕏. Furthermore, we also show that if {𝕏, 𝕐} is a regular division of a general basic configuration ℤ in ℙn, thenR/(I𝕏 + I𝕐) has the Weak Lefschetz property. These two results give positive answers to the interesting question of [39]. [ABSTRACT FROM PUBLISHER]

Stanley has conjectured that the h-vector of a matroid complex is a pure O-sequence. We will prove this for cotransversal matroids by using generalized permutohedra. We construct a bijection between lattice points inside a $r$-dimensional convex polytope and bases of a rank $r$ transversal matroid.

For any flag nestohedron, we define a flag simplicial complex whose f-vector is the γ-vector of the nestohedron. This proves that the γ-vector of any flag nestohedron satisfies the Frankl-Füredi-Kalai inequalities, partially solving a conjecture by Nevo and Petersen (Discrete Comput. Geom. 45:503-521, ). We also compare these complexes to those defined by Nevo and Petersen (Discrete Comput. Geom. 45:503-521, ) for particular flag nestohedra. [ABSTRACT FROM AUTHOR]

We study Stanley's long-standing conjecture that the h-vectors of matroid simplicial complexes are pure O-sequences. Our method consists of a new and more abstract approach, which shifts the focus from working on constructing suitable artinian level monomial ideals, as often done in the past, to the study of properties of pure O-sequences. We propose a conjecture on pure O-sequences and settle it in small socle degrees. This allows us to prove Stanley's conjecture for all matroids of rank 3. At the end of the paper, using our method, we discuss a first possible approach to Stanley's conjecture in full generality. Our technical work on pure O-sequences also uses very recent results of the third author and collaborators. [ABSTRACT FROM AUTHOR]

We consider the Ehrhart h-vector for the hypersimplex. It is well-known that the sum of the $h_{i}^{*}$ is the normalized volume which equals the Eulerian numbers. The main result is a proof of a conjecture by R. Stanley which gives an interpretation of the $h^{*}_{i}$ coefficients in terms of descents and exceedances. Our proof is geometric using a careful book-keeping of a shelling of a unimodular triangulation. We generalize this result to other closely related polytopes. [ABSTRACT FROM AUTHOR]

Let G and H be two simple graphs and let G*H denotes the graph theoretical product of G by H. In this article, we provide graded Betti numbers, Castelnuovo–Mumford regularity, projective dimension, h-vector and Hilbert series of G*H in terms of that information of G and H. More generally, we present explicit formulae to compute graded Betti numbers, h-vector, and Hilbert series of disjoint union of complexes. We will also prove that the family of graphs for which the regularity of each edge ring equals the maximum number of pairwise 3-disjoint edges, is closed under product of graphs. [ABSTRACT FROM PUBLISHER]

We introduce a construction on a flag complex that by means of modifying the associated graph generates a new flag complex whose h-vector is the face vector of the original complex. This construction yields a vertex-decomposable, hence Cohen-Macaulay, complex. From this we get a (nonnumerical) characterization of the face vectors of flag complexes and deduce also that the face vector of a flag complex is the h-vector of some vertex-decomposable flag complex. We conjecture that the converse of the latter is true and prove this, by means of an explicit construction, for hvectors of Cohen-Macaulay flag complexes arising from bipartite graphs. We also give several new characterizations of bipartite graphs with Cohen-Macaulay or Buchsbaum independence complexes. [ABSTRACT FROM AUTHOR]

Results of R. Stanley and M. Masuda completely characterize the h-vectors of simplicial posets whose order complexes are spheres. In this paper we examine the corresponding question in the case where the order complex is a ball. Using the face rings of these posets, we develop a series of new conditions on their h-vectors. We also present new methods for constructing poset balls with specific h-vectors. Combining this work with a new result of S. Murai we are able to give a complete characterization of the h-vectors of simplicial poset balls in all even dimensions, as well as odd dimensions less than or equal to five. [ABSTRACT FROM AUTHOR]

It is well known that the Stanley-Reisner ring of a matroid complex is level; this is an algebraic property between the Cohen-Macaulay and Gorenstein properties. A similar result for broken-circuit complexes is no longer true, even for graphs. We show that the Stanley-Reisner ring of the broken-circuit complex, of the cone of any simple graph, is level. [ABSTRACT FROM AUTHOR]

Abstract: The socle of a graded Buchsbaum module is studied and is related to its local cohomology modules. This algebraic result is then applied to face enumeration of Buchsbaum simplicial complexes and posets. In particular, new necessary conditions on face numbers and Betti numbers of such complexes and posets are established. These conditions are used to settle in the affirmative Kühnel''s conjecture for the maximum value of the Euler characteristic of a 2k-dimensional simplicial manifold on n vertices as well as Kalai''s conjecture providing a lower bound on the number of edges of a simplicial manifold in terms of its dimension, number of vertices, and the first Betti number. [Copyright &y& Elsevier]

Let L be a supersolvable lattice with nonzero Möbius function. We show that the order complex of any rank-selected subposet of L admits a convex-ear decomposition. This proves many new inequalities for the h-vectors of such complexes, and shows that their g-vectors are Mvectors. [ABSTRACT FROM AUTHOR]

Abstract: Let be a sequence of complex numbers such that its generating series satisfies for some polynomial . For any we study the transformation of the coefficient series of to that of where . We give a precise description of this transformation and show that under some natural mild hypotheses the roots of converge when r goes to infinity. In particular, this holds if is the Hilbert series of a standard graded k-algebra A. If in addition A is Cohen–Macaulay then the coefficients of are monotonically increasing with r. If A is the Stanley–Reisner ring of a simplicial complex Δ then this relates to the rth edgewise subdivision of Δ—a subdivision operation relevant in computational geometry and graphics—which in turn allows some corollaries on the behavior of the respective f-vectors. [Copyright &y& Elsevier]

In 1980, H. Izumi [3] introduced the concept of an h-vector. For the Finsler space whose metric is transformed by an h-vector, B. N. Prasad [10] obtained the Cartan connection. On the other hand, M. Matsumoto [7] presented a systematic theory of Finslerian hypersurface. M. Kitayama [4] obtained certain results for the Finslerian hypersurface given by β-changes. The purpose of the present paper is to derive certain properties of a Finslerian hypersurface given by an h-vector. The terminologies and notations are referred to Matsumoto [8]. [ABSTRACT FROM AUTHOR]

Abstract: Coxeter cones are formed by intersecting the nonnegative sides of a collection of root hyperplanes in some root system. They are shellable subcomplexes of the Coxeter complex, and their h-vectors record the distribution of descents among their chambers. We identify a natural class of “graded” Coxeter cones with the property that their h-vectors are symmetric and unimodal, thereby generalizing recent theorems of Reiner–Welker and Brändén about the Eulerian polynomials of graded partially ordered sets. [Copyright &y& Elsevier]

The purpose of this note is to supply an upper and a lower bound (which are in general sharp) for the h-vector of a level algebra which is relatively compressed with respect to any arbitrary level algebra A. The useful concept of relatively compressed algebra was recently introduced in Migliore et al. (2005) (whose investigations mainly focused on the particular case of A a complete intersection). The key idea of this note is the simple observation that the level algebras which are relatively compressed with respect to A coincide (after an obvious isomorphism) with the generic level quotients of suitable truncations of A. Therefore, we are able to apply to relatively compressed algebras the main result of our recent work, Zanello (2007). [ABSTRACT FROM AUTHOR]

Abstract: We show that the Ehrhart h-vector of an integer Gorenstein polytope with a regular unimodular triangulation satisfies McMullen''s g-theorem; in particular, it is unimodal. This result generalizes a recent theorem of Athanasiadis (conjectured by Stanley) for compressed polytopes. It is derived from a more general theorem on Gorenstein affine normal monoids M: one can factor (K a field) by a “long” regular sequence in such a way that the quotient is still a normal affine monoid algebra. This technique reduces all questions about the Ehrhart h-vector of P to the Ehrhart h-vector of a Gorenstein polytope Q with exactly one interior lattice point, provided each lattice point in a multiple cP, , can be written as the sum of c lattice points in P. (Up to a translation, the polytope Q belongs to the class of reflexive polytopes considered in connection with mirror symmetry.) If P has a regular unimodular triangulation, then it follows readily that the Ehrhart h-vector of P coincides with the combinatorial h-vector of the boundary complex of a simplicial polytope, and the g-theorem applies. [Copyright &y& Elsevier]

Abstract: Chari proved that if Δ is a -dimensional simplicial complex with a convex ear decomposition, then [M.K. Chari, Two decompositions in topological combinatorics with applications to matroid complexes, Trans. Amer. Math. Soc. 349 (1997) 3925–3943]. Nyman and Swartz raised the problem of whether or not the corresponding g-vector is an M-vector [K. Nyman, E. Swartz, Inequalities for h- and flag h-vectors of geometric lattices, Discrete Comput. Geom. 32 (2004) 533–548]. This is proved to be true by showing that the set of pairs , where Θ is a l.s.o.p. for , the face ring of Δ, and ω is a g-element for , is nonempty whenever the characteristic of k is zero. Finite buildings have a convex ear decomposition. These decompositions point to inequalities on the flag h-vector of such spaces similar in spirit to those examined in [K. Nyman, E. Swartz, Inequalities for h- and flag h-vectors of geometric lattices, Discrete Comput. Geom. 32 (2004) 533–548] for order complexes of geometric lattices. This also leads to connections between higher Cohen–Macaulay connectivity and conditions which insure that for a predetermined i. [Copyright &y& Elsevier]

M. Masuda recently provided the missing piece proving a conjecture of R.P. Stanley on the characterization of f-vectors for Gorenstein* simplicial posets. We propose a slight simplification of Masuda's proof. [ABSTRACT FROM AUTHOR]

Abstract: Let P be a naturally labelled, ranked (graded) poset of rank r and cardinality n. Let be the set of linear extensions of P with k descents. An explicit bijection between and is constructed using the involution principle (). A problem of Richard P. Stanley from 1981 is thereby solved. [Copyright &y& Elsevier]

We present a number of lower bounds for the h-vectors of k-Cohen-Macaulay (k-CM), broken circuit, and independence complexes. These lead to bounds on the coefficients of the characteristic and reliability polynomials of matroids. The main techniques are the use of series and parallel constructions on matroids and the short simplicial h-vector for pure complexes. [ABSTRACT FROM AUTHOR]

A difficult open problem (even in codimension 3) asks: what can be the Hilbert function of a standard graded Artinian level algebra? In this paper we solve the related problem (in codimension 3): What are the Hilbert functions H for which the minimal resolution of the lex-segment ideal with Hilbert function H permit (at least theoretically) enough cancellation to give a possible resolution for a level algebra? This gives a necessary (but not sufficient, as we show by example) response to the original open problem. The answer is given in terms of type vectors (which are equivalent to, but different from, Hilbert functions of Artinian algebras). We also give an algorithm (implemented in C.C.A) which describes how to construct all the type vectors (in codimension 3) with fixed socle degree and fixed value in that degree. [ABSTRACT FROM AUTHOR]

Assume P[sup *] is a simple, convex, d-polytope with v facets, and F[sup *] is a simple, convex d'-polytope with v' facets, where 0 < d' < d - 1. If F[sup *] is in fact a face of P[sup *] we call (P[sup *],F[sup *]) a polytope pair of type (d,v,d',v'). Define Q[sup *] to be P[sup *]F[sup *], the unbounded, simple d-polyhedron obtained by applying a projective transformation that sends a supporting hyperplane for F[sup *] onto the hyperplane at infinity. In this paper we answer the question: What are the maximum possible numbers of faces of different dimensions that P[sup *] and Q[sup *] can have? We restate and solve the problem in a dual, simplicial context. [ABSTRACT FROM AUTHOR]

More than once we have heard that the Charney-Davis Conjecture makes sense only for odd-dimensional spheres. This is to point out that in fact it is also a statement about even-dimensional spheres. [ABSTRACT FROM AUTHOR]