1,720,976 research outputs found
A numerical property of Hilbert functions and lex segment ideals
Full text linkWe introduce the fractal expansions, sequences of integers associated to a number. We show that these sequences characterize the Osequences and encode some information about lex segment ideals. Moreover, we introduce numerical functions called fractal functions, and we use them to solve the open problem of the classification of the Hilbert functions of any bigraded algebra
The minimal free resolution of fat almost complete intersections in P1 x P1
Full text linkA current research theme is to compare symbolic powers of an ideal I with the regular powers of I. In this paper, we focus on the case where I = Ix is an ideal defining an almost complete intersection (ACI) set of points X in P1 x P1. In particular, we describe a minimal free bigraded resolution of a non-arithmetically Cohen-Macaulay (also non-homogeneous) set Z of fat points whose support is an ACI, generalizing an earlier result of Cooper et al. for homogeneous sets of triple points. We call Z a fat ACI. We also show that its symbolic and ordinary powers are equal, i.e, IZ(m) = IZm for any m ≤ 1
Comparability of the total Betti numbers of toric ideals of graphs
Full text linkThe total Betti numbers of the toric ideal of a simple graph are, in general, highlysensitive to any small change of the graph. In this paper, we look at some combinatorial operations that cause total Betti numbers to change in predictable ways. In particular, we focus on a procedure that preserves these invariant
On the Betti numbers of three fat points in P1 × P1
Full text linkIn these notes we introduce a numerical function which allows us to describe explicitly (and nonrecursively) the Betti numbers, and hence, the Hilbert function of a set Z of three fat points whose support is an almost complete intersection (ACI) in P1 × P1 . A nonrecursively formula for the Betti numbers and the Hilbert function of these configurations is hard to give even for the corresponding set of five points on a special support in P2 and we did not find any kind of this result in the literature. Moreover, we also give a criterion that allows us to characterize the Hilbert functions of these special set of fat points
The ACM property for unions of lines in P1×P2
Full text linkThis paper examines the Arithmetically Cohen-Macaulay (ACM) property for certain codimension 2 varieties in P1×P2 called sets of lines in P1×P2 (not necessarily reduced). We discuss some obstacles to finding a general characterization. We then consider certain classes of such curves, and we address two questions. First, when are they themselves ACM? Second, in a non-ACM reduced configuration, is it possible to replace one component of a primary (prime) decomposition by a suitable power (i.e. to “fatten” one line) to make the resulting scheme ACM? Finally, for our classes of such curves, we characterize the locally Cohen-Macaulay property in combinatorial terms by introducing the definition of a fully v-connected configuration. We apply some of our results to give analogous ACM results for sets of lines in P3
Tower sets and other configurations with the Cohen-Macaulay property
Full text linkSome well-known arithmetically Cohen-Macaulay configurations of linear varieties in Pr as k-configurations, partial intersections and star configurations are generalized by introducing tower schemes. Tower schemes are reduced schemes that are a finite union of linear varieties whose support set is a suitable finite subset of Z+c called tower set. We prove that the tower schemes are arithmetically Cohen-Macaulay and we compute their Hilbert function in terms of their support. Afterwards, since even in codimension 2 not every arithmetically Cohen-Macaulay squarefree monomial ideal is the ideal of a tower scheme, we slightly extend this notion by defining generalized tower schemes (in codimension 2). Our main result consists in showing that the support of these configurations (the generalized tower set) gives a combinatorial characterization of the primary decomposition of the arithmetically Cohen-Macaulay squarefree monomial ideals
Linear quotients of Artinian Weak Lefschetz algebras
No full textWe study the Hilbert function and the graded Betti numbers for "generic" linear quotients of Artinian standard graded algebras, especially in the case of Weak Lefschetz algebras. Moreover, we investigate a particular property of Weak Lefschetz algebras, the Betti Weak Lefschetz Property, which makes possible to completely determine the graded Betti numbers of a generic linear quotient of such algebras
Special arrangements of lines: Codimension 2 ACM varieties in P 1 × P 1 × P 1
Full text linkIn this paper, we investigate special arrangements of lines in multiprojective spaces. In particular, we characterize codimension 2 arithmetically Cohen-Macaulay (ACM) varieties in P 1 × P 1 × P 1 , called varieties of lines. We also describe their ACM property from a combinatorial algebra point of view
Multiprojective spaces and the arithmetically Cohen–Macaulay property
Full text linkAbstractIn this paper we study the arithmetically Cohen-Macaulay (ACM) property for sets of points in multiprojective spaces. Most of what is known is for ℙ1× ℙ1and, more recently, in (ℙ1)r. In ℙ1× ℙ1the so called inclusion property characterises the ACM property. We extend the definition in any multiprojective space and we prove that the inclusion property implies the ACM property in ℙm× ℙn. In such an ambient space it is equivalent to the so-called (⋆)-property. Moreover, we start an investigation of the ACM property in ℙ1× ℙn. We give a new construction that highlights how different the behavior of the ACM property is in this setting.</jats:p
In the shadows of a hypergraph: looking for associated primes of powers of square-free monomial ideals
Full text linkThe aim of this paper is to study the associated primes of powers of square-free monomial ideals. Each square-free monomial ideal corresponds uniquely to a finite simple hypergraph via the cover ideal construction, and vice versa. Let H be a finite simple hypergraph and J(H) the cover ideal of H. We define the shadows of hypergraph, H, described as a collection of smaller hypergraphs related to H under some conditions. We then investigate how the shadows of H preserve information about the associated primes of the powers of J(H). Finally, we apply our findings on shadows to study the persistence property of square-free monomial ideals and construct some examples exhibiting failure of containment
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