We study the arithmetically Cohen-Macaulay (ACM) property for finite sets of points in multiprojective spaces, especially (P 1 ) n . A combinatorial characterization, the (⋆)-property, is known in P 1 × P 1 . We propose a combinatorial property, (⋆ s ) with 2 ≤ s ≤ n, that directly generalizes the (⋆)-property to (P 1 ) n for larger n. We show that X is ACM if and only if it satisfies the (⋆ n )-property. The main tool for several of our results is an extension to the multiprojective setting of certain liaison methods in projective space.2010 Mathematics Subject Classification. 13C40, 13C14, 13A15, 14M05.
The aim of this paper is to make a connection between design theory and algebraic geometry/commutative algebra. In particular, given any Steiner System S(t, n, v) we associate two ideals, in a suitable polynomial ring, defining a Steiner configuration of points and its Complement. We focus on the latter, studying its homological invariants, such as Hilbert Function and Betti numbers. We also study symbolic and regular powers associated to the ideal defining a Complement of a Steiner configuration of points, finding its Waldschmidt constant, regularity, bounds on its resurgence and asymptotic resurgence. We also compute the parameters of linear codes associated to any Steiner configuration of points and its Complement.
In this paper we study the arithmetically Cohen-Macaulay (ACM) property for sets of points in multiprojective spaces. Most of what is known is for P 1 × P 1 and, more recently, in (P 1 ) r . In P 1 × P 1 the so called inclusion property characterizes the ACM property. We extend the definition in any multiprojective space and we prove that the inclusion property implies the ACM property in P m × P n . In such an ambient space it is equivalent to the so-called (⋆)-property. Moreover, we start an investigation of the ACM property in P 1 × P n . We give a new construction that highlights how different the behavior of the ACM property is in this setting.2010 Mathematics Subject Classification. 14M05, 13C14, 13C40, 13H10, 13A15.
Abstract. A 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 that I = I X is an ideal defining an almost complete intersection (ACI) set of points X in P 1 × P 1 . In particular, we describe a minimal free bigraded resolution of a non arithmetically Cohen-Macaulay (also non homogeneus) set Z of fat points whose support is an ACI generalizing Corollary 4.6 given in [5] 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, I (m) Z = I m Z for any m ≥ 1.
For all integers
4
≤
r
≤
d
4 \leq r \leq d
, we show that there exists a finite simple graph
G
=
G
r
,
d
G= G_{r,d}
with toric ideal
I
G
⊂
R
I_G \subset R
such that
R
/
I
G
R/I_G
has (Castelnuovo–Mumford) regularity
r
r
and
h
h
-polynomial of degree
d
d
. To achieve this goal, we identify a family of graphs such that the graded Betti numbers of the associated toric ideal agree with its initial ideal, and, furthermore, that this initial ideal has linear quotients. As a corollary, we can recover a result of Hibi, Higashitani, Kimura, and O’Keefe that compares the depth and dimension of toric ideals of graphs.
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