1,412 research outputs found
The Hilbert functions of ACM sets of points in P^{n_1} x ... x P^{n_k}
The Hilbert functions of sets of distinct points in P^n have been
characterized. We show that if we restrict to sets of distinct of points in
P^{n_1} x ... x P^{n_k} that are also arithmetically Cohen-Macaulay (ACM for
short), then there is a natural generalization of this result. We begin by
determining the possible values for the invariants K-dim R/Ix and depth R/Ix,
where R/Ix is the coordinate ring associated to a set of distinct points X in
P^{n_1} x ... x P^{n_k}. At the end of this paper we give a new
characterization of ACM sets of points in P^1 x P^1.Comment: 18 pages, v.2 to appear in J. Alg. Sections 2 and 3 have been
combined, Section 5 has been removed, and Section 6 has been split into two
section
Simplicial complexes and Macaulay's inverse systems
Let be a simplicial complex on , with
Stanley-Reisner ideal . The goal of
this paper is to investigate the class of artinian algebras
, where each
. By utilizing the technique of Macaulay's inverse systems, we can
explicitly describe the socle of in terms of . As a consequence, we
determine the simplicial complexes, that we will call {\em levelable}, for
which there exists a tuple such that is
a level algebra.Comment: Very minor changes. To appear in Math.
ACM sets of points in multiprojective space
If X is a finite set of points in a multiprojective space P^n1 x ... x P^nr
with r >= 2, then X may or may not be arithmetically Cohen-Macaulay (ACM). For
sets of points in P^1 x P^1 there are several classifications of the ACM sets
of points. In this paper we investigate the natural generalizations of these
classifications to an arbitrary multiprojective space. We show that each
classification for ACM points in P^1 x P^1 fails to extend to the general case.
We also give some new necessary and sufficient conditions for a set of points
to be ACM.Comment: 21 pages; revised final version; minor corrections; to appear in
Collectanea Mathematic
Star configuration points and plane curves
2siLet ℓ1,...,ℓ1 be l lines in ℙ2 such that no three lines meet in a point. Let X(l) be the set of points {ℓi ∩ ℓj {divides} 1 ≤ i < j ≤ l} ⊆ ℙ2. We call X(l) a star configuration. We describe all pairs (d, l) such that the generic degree d curve in ℙ2 contains an X(l). Our proof strategy uses both a theoretical and an explicit algorithmic approach. We also describe how one may extend our algorithmic approach to similar problems. © 2011 American Mathematical Society.openopenCarlini E.; van Tuyl A.Carlini, E.; van Tuyl, A
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