On the configurations of codimension two linear subspaces of ℙN with the Waldschmidt constant less than 2

2020 ◽  
pp. 2150178
Author(s):  
Hassan Haghighi ◽  
Mohammad Mosakhani
Keyword(s):  
Hilbert Function ◽  
Free Resolution ◽  
Minimal Free Resolution ◽  
Codimension Two ◽  
Symbolic Powers ◽  
Point Configurations ◽  
Higher Dimensional ◽  
Algebraic Invariants ◽  
Waldschmidt Constants ◽  
Waldschmidt Constant

The purpose of this note is to generalize a result of [M. Dumnicki, T. Szemberg and H. Tutaj-Gasińska, Symbolic powers of planar point configurations II, J. Pure Appl. Alg. 220 (2016) 2001–2016] to higher-dimensional projective spaces and classify all configurations of [Formula: see text]-planes [Formula: see text] in [Formula: see text] with the Waldschmidt constants less than two. We also determine some numerical and algebraic invariants of the defining ideals [Formula: see text] of these classes of configurations, i.e. the resurgence, the minimal free resolution and the regularity of [Formula: see text], as well as the Hilbert function of [Formula: see text].

scholarly journals Initial sequences and Waldschmidt constants of planar point configurations

2017 ◽  
Vol 27 (06) ◽  
pp. 717-729 ◽  
Author(s):  
Łucja Farnik ◽  
J. Gwoździewicz ◽  
B. Hejmej ◽  
M. Lampa-Baczyńska ◽  
G. Malara ◽  
...  
Keyword(s):  
Planar Point ◽  
Symbolic Powers ◽  
Point Configurations ◽  
Waldschmidt Constants ◽  
Waldschmidt Constant

The purpose of this work is to extend the classification of planar point configurations with low Waldschmidt constants initiated in [M. Dumnicki, T. Szemberg and H. Tutaj-Gasińska, Symbolic powers of planar point configurations II, J. Pure Appl. Algebra 220 (2016) 2001–2016] and continued in [M. Mosakhani and H. Haghighi, On the configurations of points in [Formula: see text] with the Waldschmidt constant equal to two, J. Pure Appl. Algebra 220 (2016) 3821–3825] for all values less than [Formula: see text]. As a consequence, we prove a conjecture of Dumnicki, Szemberg and Tutaj-Gasińska concerning initial sequences with low first differences.

Fat Points in ℙ1× ℙ1and Their Hilbert Functions

2004 ◽  
Vol 56 (4) ◽  
pp. 716-741 ◽  
Author(s):  
Elena Guardo ◽  
Adam Van Tuyl
Keyword(s):  
Finite Number ◽  
Hilbert Function ◽  
Free Resolution ◽  
Hilbert Functions ◽  
Minimal Free Resolution ◽  
Fat Points ◽  
Fat Point ◽  
Point Scheme

AbstractWe study the Hilbert functions of fat points in ℙ1× ℙ1. IfZ⊆ ℙ1× ℙ1is an arbitrary fat point scheme, then it can be shown that for everyiandjthe values of the Hilbert functionHZ(l,j) andHZ(i,l) eventually become constant forl≫ 0. We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of the points in ℙ1× ℙ1. This enables us to compute all but a finite number values ofHZwithout using the coordinates of points. We also characterize the ACM fat point schemes using our description of the eventual behaviour. In fact, in the case thatZ⊆ ℙ1× ℙ1is ACM, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.

The Minimal Free Resolution of a Fat Star-Configuration in ℙn

2014 ◽  
Vol 21 (01) ◽  
pp. 157-166 ◽  
Author(s):  
Jeaman Ahn ◽  
Yong Su Shin
Keyword(s):  
Hilbert Function ◽  
Free Resolution ◽  
Minimal Free Resolution ◽  
Star Configuration

We find the minimal free resolution of a fat star-configuration 𝕏 in ℙn of type (r,s,t) defined by general forms of degrees d1, …, dr, and show that a fat linear star-configuration 𝕏 in ℙ2 never has generic Hilbert function if (s,t) ≠ (1,1) or (2,2). These two results generalize the interesting results of [2].

The Minimal Free Resolution of Fat Almost Complete Intersections in ℙ1 × ℙ1

2017 ◽  
Vol 69 (6) ◽  
pp. 1274-1291 ◽  
Author(s):  
Giuseppe Favacchio ◽  
Elena Guardo
Keyword(s):  
Complete Intersection ◽  
Free Resolution ◽  
Complete Intersections ◽  
Minimal Free Resolution ◽  
Symbolic Powers ◽  
Research Theme ◽  
Homogeneous Set ◽  
Triple Points ◽  
Set Of Points ◽  
A Current

AbstractA 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 deûning an almost complete intersection (ACI) set of points X in ℙ1 × 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, .

THE PROJECTIVE DIMENSION OF THE EDGE IDEAL OF A VERY WELL-COVERED GRAPH

2017 ◽  
Vol 230 ◽  
pp. 160-179 ◽  
Author(s):  
KYOUKO KIMURA ◽  
NAOKI TERAI ◽  
SIAMAK YASSEMI
Keyword(s):  
Projective Dimension ◽  
Free Resolution ◽  
Covering Number ◽  
Edge Ideal ◽  
Minimal Free Resolution ◽  
Symbolic Powers

A very well-covered graph is an unmixed graph whose covering number is half of the number of vertices. We construct an explicit minimal free resolution of the cover ideal of a Cohen–Macaulay very well-covered graph. Using this resolution, we characterize the projective dimension of the edge ideal of a very well-covered graph in terms of a pairwise$3$-disjoint set of complete bipartite subgraphs of the graph. We also show nondecreasing property of the projective dimension of symbolic powers of the edge ideal of a very well-covered graph with respect to the exponents.

The hilbert function and the minimal free resolution of some gorenstein ideals of codimension 4

1998 ◽  
Vol 26 (12) ◽  
pp. 4285-4307 ◽  
Author(s):  
Anthony V. Geramita ◽  
Hyoung June Ko ◽  
Yong Su Shin
Keyword(s):  
Hilbert Function ◽  
Free Resolution ◽  
Minimal Free Resolution

Linear Syzygies from k-Configurations in ℙn and the Mapping Cone Constructions

2007 ◽  
Vol 14 (04) ◽  
pp. 649-660
Author(s):  
Yong Su Shin
Keyword(s):  
Hilbert Function ◽  
Free Resolution ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Minimal Free Resolution ◽  
Mapping Cone ◽  
Linear Syzygies ◽  
Necessary And Sufficient

We find a necessary and sufficient condition for the Hilbert function to support the ith linear syzygy using type vectors and also construct an Artinian level algebra whose minimal free resolution supports the nth linear syzygy using both a mapping cone construction and coordinate points in ℙn.

scholarly journals Waldschmidt constants for Stanley–Reisner ideals of a class of simplicial complexes

2016 ◽  
Vol 15 (07) ◽  
pp. 1650137 ◽  
Author(s):  
Cristiano Bocci ◽  
Barbara Franci
Keyword(s):  
Simplicial Complexes ◽  
Combinatorial Approach ◽  
Ideal Formula ◽  
Symbolic Powers ◽  
Waldschmidt Constants ◽  
Waldschmidt Constant

We study the symbolic powers of the Stanley–Reisner ideal [Formula: see text] of a bipyramid [Formula: see text] over a [Formula: see text]-gon [Formula: see text]. Using a combinatorial approach, based on analysis of subtrees in [Formula: see text] we compute the Waldschmidt constant of [Formula: see text].

scholarly journals Curves with Two Pencils and Associated Maps to Projective Spaces

2006 ◽  
Vol 13 (3) ◽  
pp. 411-417
Author(s):  
Edoardo Ballico
Keyword(s):  
Linear System ◽  
Projective Spaces ◽  
Free Resolution ◽  
Homogeneous Ideal ◽  
Projective Curve ◽  
Minimal Free Resolution ◽  
Complete Linear System

Abstract Let 𝑋 be a smooth and connected projective curve. Assume the existence of spanned 𝐿 ∈ Pic𝑎(𝑋), 𝑅 ∈ Pic𝑏(𝑋) such that ℎ0(𝑋, 𝐿) = ℎ0(𝑋, 𝑅) = 2 and the induced map ϕ 𝐿,𝑅 : 𝑋 → 𝐏1 × 𝐏1 is birational onto its image. Here we study the following question. What can be said about the morphisms β : 𝑋 → 𝐏𝑅 induced by a complete linear system |𝐿⊗𝑢⊗𝑅⊗𝑣| for some positive 𝑢, 𝑣? We study the homogeneous ideal and the minimal free resolution of the curve β(𝑋).

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