Reducible family of height three level algebras

Let $R=k[x_1,..., x_r]$ be the polynomial ring in $r$ variables over an infinite field $k$, and let $M$ be the maximal ideal of $R$. Here a \emph{level algebra} will be a graded Artinian quotient $A$ of $R$ having socle $Soc(A)=0:M$ in a single degree $j$. The Hilbert function $H(A)=(h_0,h_1,... ,h_j)$ gives the dimension $h_i=\dim_k A_i$ of each degree-$i$ graded piece of $A$ for $0\le i\le j$. The embedding dimension of $A$ is $h_1$, and the \emph{type} of $A$ is \dim_k \Soc (A), here $h_j$. The family \Levalg (H) of level algebra quotients of $R$ having Hilbert function $H$ forms an open subscheme of the family of graded algebras or, via Macaulay duality, of a Grassmannian. We show that for each of the Hilbert functions $H=H_1=(1,3,4,4)$ and $H=H_2=(1,3,6,8,9,3)$ the family $LevAlg (H)$ parametrizing level Artinian algebras of Hilbert function $H$ has several irreducible components. We show also that these examples each lift to points. However, in the first example, an irreducible Betti stratum for Artinian algebras becomes reducible when lifted to points. These were the first examples we obtained of multiple components for \Levalg(H) in embedding dimension three. We also show that the second example is the first in an infinite sequence of examples of type three Hilbert functions $H(c)$ in which also the number of components of LevAlg(H) gets arbitrarily large. The first case where the phenomenon of multiple components can occur (i.e. the lowest embedding dimension and then the lowest type) is that of dimension three and type two. Examples of this first case have been obtained by the authors and also by J.-O. Kleppe.Comment: 20 pages. Minor revisio

Graded Betti numbers of Cohen-Macaulay modules and the Multiplicity conjecture

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.Comment: 24 pages, references and examples adde

Some algebraic consequences of Green's hyperplane restriction theorems

We discuss a paper of M. Green from a new algebraic perspective, and provide applications of its results to level and Gorenstein algebras, concerning their Hilbert functions and the weak Lefschetz property. In particular, we will determine a new infinite class of symmetric $h$-vectors that cannot be Gorenstein $h$-vectors, which was left open in a recent work of Migliore-Nagel-Zanello. This includes the smallest example previously unknown, $h=(1,10,9,10,1)$. As M. Green's results depend heavily on the characteristic of the base field, so will ours. The appendix will contain a new argument, kindly provided to us by M. Green, for Theorems 3 and 4 of his paper, since we had found a gap in the original proof of those results during the preparation of this manuscript.Comment: A few revisions. Final version to appear in JPA

Cones of Hilbert functions

We study the closed convex hull of various collections of Hilbert functions. Working over a standard graded polynomial ring with modules that are generated in degree zero, we describe the supporting hyperplanes and extreme rays for the cones generated by the Hilbert functions of all modules, all modules with bounded a-invariant, and all modules with bounded Castelnuovo-Mumford regularity. The first of these cones is infinite-dimensional and simplicial, the second is finite-dimensional but neither simplicial nor polyhedral, and the third is finite-dimensional and simplicial.Comment: 20 pages, 2 figure

A classification of the weak Lefschetz property for almost complete intersections generated by uniform powers of general linear forms

We settle a conjecture by Migliore, Mir\'o-Roig, and Nagel which gives a classification of the Weak Lefschetz property for almost complete intersections generated by uniform powers of general linear forms.Comment: 13 page