Projective curves of degree=codimension+2 II

2016 ◽  
Vol 26 (01) ◽  
pp. 95-104 ◽  
Author(s):  
Wanseok Lee ◽  
Euisung Park
Keyword(s):  
Integral Curve ◽  
Betti Numbers ◽  
Free Resolution ◽  
Relative Location ◽  
Normal Curve ◽  
Minimal Free Resolution ◽  
Graded Betti Numbers ◽  
Projective Curves ◽  
Rational Normal Curve

Let [Formula: see text] be a nondegenerate projective integral curve of degree [Formula: see text] which is not linearly normal. In this paper, we continues the study begun in [E. Park, Projective curves of degree=codimension+2, Math. Z. 256 (2007) 685–697] for the minimal free resolution of [Formula: see text]. It is well-known that [Formula: see text] is an isomorphic projection of a rational normal curve [Formula: see text] from a point [Formula: see text]. Our main result is about how the graded Betti numbers of [Formula: see text] are determined by the rank of [Formula: see text] with respect to [Formula: see text], which is a measure of the relative location of [Formula: see text] from [Formula: see text].

Multi-graded Betti numbers of path ideals of trees

2017 ◽  
Vol 16 (01) ◽  
pp. 1750018 ◽  
Author(s):  
Rachelle R. Bouchat ◽  
Tricia Muldoon Brown
Keyword(s):  
Betti Numbers ◽  
Free Resolution ◽  
Explicit Description ◽  
Free Resolutions ◽  
Minimal Free Resolution ◽  
Generating Set ◽  
Graded Betti Numbers ◽  
Minimal Free Resolutions ◽  
Vertex Sets ◽  
Minimal Generating Set

A path ideal of a tree is an ideal whose minimal generating set corresponds to paths of a specified length in a tree. We provide a description of a collection of induced subtrees whose vertex sets correspond to the multi-graded Betti numbers on the linear strand in the corresponding minimal free resolution of the path ideal. For two classes of path ideals, we give an explicit description of a collection of induced subforests whose vertex sets correspond to the multi-graded Betti numbers in the corresponding minimal free resolutions. Lastly, in both classes of path ideals considered, the graded Betti numbers are explicitly computed for [Formula: see text]-ary trees.

scholarly journals From grids to pseudo-grids of lines: Resolution and seminormality

2020 ◽  
pp. 2150213
Author(s):  
Francesca Cioffi ◽  
Margherita Guida ◽  
Luciana Ramella
Keyword(s):  
Analogous Result ◽  
Betti Numbers ◽  
Free Resolution ◽  
Infinite Field ◽  
Minimal Free Resolution ◽  
Mapping Cone ◽  
Configurations Of Lines

Over an infinite field [Formula: see text], we investigate the minimal free resolution of some configurations of lines. We explicitly describe the minimal free resolution of complete grids of lines and obtain an analogous result about the so-called complete pseudo-grids. Moreover, we characterize the total Betti numbers of configurations that are obtained posing a multiplicity condition on the lines of either a complete grid or a complete pseudo-grid. Finally, we analyze when a complete pseudo-grid is seminormal, differently from a complete grid. The main tools that have been involved in our study are the mapping cone procedure and properties of liftings, of pseudo-liftings and of weighted ideals. Although complete grids and pseudo-grids are hypersurface configurations and many results about such type of configurations have already been stated in literature, we give new contributions, in particular about the maps of the resolution.

scholarly journals On the Betti numbers and Rees algebras of ideals with linear powers

2021 ◽  
Vol 53 (2) ◽  
pp. 575-592
Author(s):  
Lisa Nicklasson
Keyword(s):  
Betti Numbers ◽  
Free Resolution ◽  
Minimal Free Resolution ◽  
Linear Relations ◽  
Rees Algebras ◽  
Low Dimension ◽  
The Ideal

AbstractAn ideal $$I \subset \mathbb {k}[x_1, \ldots , x_n]$$ I ⊂ k [ x 1 , … , x n ] is said to have linear powers if $$I^k$$ I k has a linear minimal free resolution, for all integers $$k>0$$ k > 0 . In this paper, we study the Betti numbers of $$I^k$$ I k , for ideals I with linear powers. We provide linear relations on the Betti numbers, which holds for all ideals with linear powers. This is especially useful for ideals of low dimension. The Betti numbers are computed explicitly, as polynomials in k, for the ideal generated by all square-free monomials of degree d, for $$d=2, 3$$ d = 2 , 3 or $$n-1$$ n - 1 , and the product of all ideals generated by s variables, for $$s=n-1$$ s = n - 1 or $$n-2$$ n - 2 . We also study the generators of the Rees ideal, for ideals with linear powers. Particularly, we are interested in ideals for which the Rees ideal is generated by quadratic elements. This problem is related to a conjecture on matroids by White.

scholarly journals Betti Numbers of Transversal Monomial Ideals

2011 ◽  
Vol 18 (spec01) ◽  
pp. 925-936
Author(s):  
Rahim Zaare-Nahandi
Keyword(s):  
Monomial Ideal ◽  
Betti Numbers ◽  
Free Resolution ◽  
Circulant Matrix ◽  
Monomial Ideals ◽  
Minimal Free Resolution ◽  
Ground Field ◽  
Initial Ideal ◽  
The Matrix ◽  
Generic Singularities

In this paper, by a modification of a previously constructed minimal free resolution for a transversal monomial ideal, the Betti numbers of this ideal is explicitly computed. For convenient characteristics of the ground field, up to a change of coordinates, the ideal of t-minors of a generic pluri-circulant matrix is a transversal monomial ideal. Using a Gröbner basis for this ideal, it is shown that the initial ideal of a generic pluri-circulant matrix is a stable monomial ideal when the matrix has two square blocks. By means of the Eliahou-Kervaire resolution for stable monomial ideals, the Betti numbers of this initial ideal is computed and it is proved that for some significant values of t, this ideal has the same Betti numbers as the corresponding transversal monomial ideal. The ideals treated in this paper naturally arise in the study of generic singularities of algebraic varieties.

scholarly journals Minimal free resolutions of 2 × n domino tilings

2019 ◽  
Vol 18 (06) ◽  
pp. 1950118
Author(s):  
Rachelle R. Bouchat ◽  
Tricia Muldoon Brown
Keyword(s):  
Monomial Ideal ◽  
Betti Numbers ◽  
Projective Dimension ◽  
Free Resolution ◽  
Free Resolutions ◽  
Minimal Free Resolution ◽  
Domino Tilings ◽  
Minimal Free Resolutions ◽  
The Ideal

We introduce a squarefree monomial ideal associated to the set of domino tilings of a [Formula: see text] rectangle and proceed to study the associated minimal free resolution. In this paper, we use results of Dalili and Kummini to show that the Betti numbers of the ideal are independent of the underlying characteristic of the field, and apply a natural splitting to explicitly determine the projective dimension and Castelnuovo–Mumford regularity of the ideal.

scholarly journals Betti Numbers of Monomial Ideals and Shifted Skew Shapes

10.37236/69 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Uwe Nagel ◽  
Victor Reiner
Keyword(s):  
Lower Bound ◽  
Betti Numbers ◽  
Free Resolution ◽  
Monomial Ideals ◽  
Fixed Degree ◽  
Minimal Free Resolution ◽  
Edge Ideals ◽  
Combinatorial Interpretations ◽  
Ferrers Graphs ◽  
Strongly Stable Ideals

We present two new problems on lower bounds for Betti numbers of the minimal free resolution for monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are quadratic and bihomogeneous with respect to two variable sets, but gives a more finely graded lower bound. These problems are solved for certain classes of ideals that generalize (in two different directions) the edge ideals of threshold graphs and Ferrers graphs. In the process, we produce particularly simple cellular linear resolutions for strongly stable and squarefree strongly stable ideals generated in a fixed degree, and combinatorial interpretations for the Betti numbers of other classes of ideals, all of which are independent of the coefficient field.

scholarly journals Homology of Powers of Ideals: Artin-Rees Numbers of Syzygies and the Golod Property

2016 ◽  
Vol 23 (04) ◽  
pp. 689-700 ◽  
Author(s):  
Jürgen Herzog ◽  
Volkmar Welker ◽  
Siamak Yassemi
Keyword(s):  
Polynomial Ring ◽  
Residue Class ◽  
Betti Numbers ◽  
Free Resolution ◽  
Minimal Free Resolution ◽  
Powers Of Ideals ◽  
Residue Class Field ◽  
Syzygy Modules ◽  
Free Modules ◽  
Specific Degree

Let R0 be a Noetherian local ring and R a standard graded R0-algebra with maximal ideal 𝔪 and residue class field 𝕂 = R/𝔪. For a graded ideal I in R we show that for k ≫ 0: (1) the Artin-Rees number of the syzygy modules of Ik as submodules of the free modules from a free resolution is constant, and thereby the Artin-Rees number can be presented as a proper replacement of regularity in the local situation; and (2) R is a polynomial ring over the regular R0, the ring R/Ik is Golod, its Poincaré-Betti series is rational and the Betti numbers of the free resolution of 𝕂 over R/Ik are polynomials in k of a specific degree. The first result is an extension of the work by Swanson on the regularity of Ik for k ≫ 0 from the graded situation to the local situation. The polynomiality consequence of the second result is an analog of the work by Kodiyalam on the behaviour of Betti numbers of the minimal free resolution of R/Ik over R.

scholarly journals Betti numbers of Bresinsky’s curves in 𝔸4

2019 ◽  
Vol 18 (08) ◽  
pp. 1950143 ◽  
Author(s):  
Ranjana Mehta ◽  
Joydip Saha ◽  
Indranath Sengupta
Keyword(s):  
Betti Number ◽  
Betti Numbers ◽  
Free Resolution ◽  
Minimal Free Resolution ◽  
Monomial Curves ◽  
Number Of Generators

Bresinsky defined a class of monomial curves in [Formula: see text] with the property that the minimal number of generators or the first Betti number of the defining ideal is unbounded above. We prove that the same behavior of unboundedness is true for all the Betti numbers and construct an explicit minimal free resolution for the defining ideal of this class of curves.

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).

Sign in / Sign up

Export Citation Format

Share Document