Abstract. Let A = ⊕ i≥0 Ai be a standard graded Artinian K-algebra, where char K = 0. Then A has the Weak Lefschetz property if there is an element ℓ of degree 1 such that the multiplication ×ℓ: Ai → Ai+1 has maximal rank, for every i, and A has the Strong Lefschetz property if ×ℓ d: Ai → Ai+d has maximal rank for every i and d. The main results obtained in this paper are the following. 1. Every height three complete intersection has the Weak Lefschetz property. (Our method, surprisingly, uses rank two vector bundles on P 2 and the Grauert-Mülich theorem.) 2. We give a complete characterization (including a concrete construction) of the Hilbert functions that can occur for K-algebras with the Weak or Strong Lefschetz property (and the characterization is the same one!). 3. We give a sharp bound on the graded Betti numbers (achieved by our construction) of Artinian K-algebras with the Weak or Strong Lefschetz property and fixed Hilbert function. This bound is again the same for both properties! Some Hilbert functions in fact force the algebra to have the maximal Betti numbers. 4. Every Artinian ideal in K[x, y] possesses the Strong Lefschetz property. This is false in higher codimension. 1.

Abstract. Let d1,..., dr be positive integers and let I = (F1,..., Fr) be an ideal generated by forms of degrees d1,...,dr, respectively, in a polynomial ring R with n variables. With no further information virtually nothing can be said about I, even if we add the assumption that R/I is Artinian. Our first object of study is the case where the Fi are chosen generally, subject only to the degree condition. When all the degrees are the same we give a result that says, roughly, that they have as few first syzygies as possible. In the general case, the Hilbert function of R/I has been conjectured by Fröberg. In a previous work the authors showed that in many situations the minimal free resolution of R/I must have redundant terms which are not forced by Koszul (first or higher) syzygies among the Fi (and hence could not be predicted from the Hilbert function), but the only examples came when r = n + 1. Our second main set of results in this paper show that examples can be obtained when n + 1 ≤ r ≤ 2n − 2. Finally, we show that if Fröberg’s conjecture on the Hilbert function is true then any such redundant terms in the minimal free resolution must occur in the top two possible degrees of the free module. Closely connected to the Fröberg conjecture is the notion of Strong Lefschetz property, and slightly less closely connected is the Weak Lefschetz property. We also study an intermediate notion, called the Maximal Rank property. We continue the description of the ubiquity of these properties, especially the Weak Lefschetz property. We show that any ideal of general forms in k[x1, x2, x3, x4] has the Weak Lefschetz property. Then we show that for certain choices of degrees, any complete intersection has the Weak Lefschetz property and any almost complete intersection has the Weak Lefschetz property. Finally, we show that most of the time Artinian “hypersurface sections ” of zeroschemes have the Weak Lefschetz property. Contents

Abstract. Let R = k[x1,..., xn] and let I be the ideal of n + 1 generically chosen forms of degrees d1 ≤ · · · ≤ dn+1. We give the precise graded Betti numbers of R/I in the following cases: • n = 3. • n = 4 and ∑5 i=1 di is even. • n = 4, ∑5

We give conjectures on the possible graded Betti numbers of Cohen-Macaulay modules up to multiplication by positive rational numbers. The idea is that the Betti diagrams should be non-negative linear combinations of pure diagrams. The conjectures are verified in the cases where the structure of resolutions are known, i.e., for modules of codimension two, for Gorenstein algebras of codimension three and for complete intersections. The motivation for the conjectures comes from the Multiplicity conjecture of Herzog, Huneke and Srinivasan.

INTRODUCTION The k-dimensional skeleton of a d-polytope P is the set of all faces of the polytope of dimension at most k. The 1-skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent if they form the endpoints of an edge of P . In this chapter, we will describe results and problems concerning graphs and skeletons of polytopes. In Section 17.1 we briefly describe the situation for 3polytopes. In Section 17.2 we consider general properties of polytopal graphs--- subgraphs and induced subgraphs, connectivity and separation, expansion, and other properties. In Section 17.3 we discuss problems related to diameters of polytopal graphs in connection with the simplex algorithm and t

ABSTRACT. In this note we give a new (and rather simple) proof of the following well-known theorem of R. Stanley: the h-vectors of Gorenstein algebras of codimension 3 are SI-sequences, i.e. are symmetric and the first difference of their first half is an O-sequence. We consider standard graded artinian algebras A = R/I, where R = k[x1,..., xr], k is any field, the xi’s have degree 1 and I is a homogeneous ideal of R. Recall that the h-vector of A is h(A) = h = (h0, h1,..., he), where hi = dimk Ai and e is the last index such that dimk Ae> 0. Since we may suppose that I does not contain non-zero forms of degree 1, r = h1 is defined to be the codimension of A. The socle of A is the annihilator of the maximal homogeneous ideal m = (x1,..., xr) ⊂ A, namely soc(A) = {a ∈ A | am = 0}. Since soc(A) is a homogeneous ideal, we define the socle-vector of A as s(A) = s = (s0, s1,..., se), where si = dimk soc(A)i. Note that se = he> 0. The integer e is called the socle degree of A (or of h(A)). If s = (0, 0,..., 0, se = 1), we say that the algebra A is Gorenstein. The next theorem is a well-known result of Macaulay. Definition-Remark 1. Let n and i be positive integers. The i-binomial expansion of 1 n is n(i) = where ni> ni−1>...> nj ≥ j ≥ 1. ni

Abstract. The conjecture of Kalai, Kleinschmidt, and Lee on the number of empty simplices of a simplicial polytope is established by relating it to the first graded Betti numbers of the polytope. The proof allows us to derive explicit optimal bounds on the number of empty simplices of any given dimension. As a key result, we prove optimal bounds for the graded Betti numbers of any standard graded K-algebra in terms of its Hilbert function. 1.

These are the expanded and detailed notes of the lectures given by the authors during the school and workshop entitled “Liaison and related topics”, held at the Politecnico of Torino during the period October 1-5, 2001. In the notes we have attempted to cover liaison theory from first principles, through the main developments (especially in codimension two) and the standard applications, to the recent developments in Gorenstein liaison and a discussion of open problems. Given the extensiveness of the subject, it was not possible to go into great detail in every proof. Still, it is hoped that the material that we chose will be beneficial and illuminating for the principants, and for the reader.

Abstract. The main goal of this paper is to characterize the Hilbert functions of all (artinian) codimension 4 Gorenstein algebras that have at least two independent relations of degree four. This includes all codimension 4 Gorenstein algebras whose initial relation is of degree at most 3. Our result shows that those Hilbert functions are exactly the so-called SI-sequences starting with (1, 4, h2, h3,...), where h4 ≤ 33. In particular, these Hilbert functions are all unimodal. We also establish a more general unimodality result, which relies on the values of the Hilbert function not being too big, but is independent of the initial degree. 1.

In this paper we study standard graded artinian level algebras, in particular those whose socle-vector has type 2. Our main results are: the characterization of the level h-vectors of the form (1, r,..., r, 2) for r ≤ 4; the characterization of the minimal free resolutions associated to each of the h-vectors above when r = 3; a sharp upperbound (under certain mild hypotheses) for the level h-vectors (1, r,..., a, 2) of arbitrary codimension r and type 2, which depends on the next to last entry a.