Powers of Square-Free Monomial Ideals and Combinatorics

Francisco, Christopher A.; Harbourne, Brian; Mermin, Jeffrey

doi:10.1007/978-1-4614-5292-8_11

Cited by 13 publications

(11 citation statements)

References 26 publications

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“…If a vertex x i of D is a source (i.e., has only arrows leaving x i ) we shall always assume d i = 1 because in this case the definition of I(D) does not depend on the weight of x i . In the special case when d i = 1 for all i, we recover the edge ideal of the graph G which has been extensively studied in the literature [8,11,14,18,20,30,38,39,40,42]. A vertex-weighted digraph D is called Cohen-Macaulay (over the field K) if R/I(D) is a Cohen-Macaulay ring.…”

Section: Introductionmentioning

confidence: 99%

Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs

Gimenez

Martinez-Bernal

Simis

et al. 2018

Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics

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In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertexweighted oriented graphs. These are digraphs having no oriented cycles of length two with weights on the vertices. For a monomial ideal with no embedded primes we classify the normality of its symbolic Rees algebra in terms of its primary components. If the primary components of a monomial ideal are normal, we present a simple procedure to compute its symbolic Rees algebra using Hilbert bases, and give necessary and sufficient conditions for the equality between its ordinary and symbolic powers. We give an effective characterization of the Cohen-Macaulay vertexweighted oriented forests. For edge ideals of transitive weighted oriented graphs we show that Alexander duality holds. It is shown that edge ideals of weighted acyclic tournaments are Cohen-Macaulay and satisfy Alexander duality.

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Section: Introductionmentioning

confidence: 99%

Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs

Gimenez

Martinez-Bernal

Simis

et al. 2018

Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics

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“…In particular, according to Proposition 2.7 in [6], one has J(H) = I(H) ∨ , where I(H) ∨ denotes the Alexander dual of I(H).…”

Section: Preliminariesmentioning

confidence: 99%

Classes of normally and nearly normally torsion-free monomial ideals

Nasernejad,

Qureshi,

Khashyarmanesh

et al. 2021

Preprint

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“…We begin by recalling briefly basic notions from combinatorial algebra relevant in this note, for more detailed account see the very nice surveys [6] and [7]. The Stanley-Reisner ideals introduced here have traditionally provided a rich source of non-trivial examples.…”

Section: Bipyramids Revisitedmentioning

confidence: 99%

Waldschmidt Constants for Stanley–Reisner Ideals of a Class of Graphs

Szemberg

Szpond

2018

Springer Proceedings in Mathematics &Amp; Statistics

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In the present note we study Waldschmidt constants of Stanley-Reisner ideals of a hypergraph and a graph with vertices forming a bipyramid over a planar n-gon. The case of the hypergraph has been studied by Bocci and Franci. We reprove their main result. The case of the graph is new. Interestingly, both cases provide series of ideals with Waldschmidt constants descending to 1. It would be interesting to known if there are bounded ascending sequences of Waldschmidt constants.

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Powers of Square-Free Monomial Ideals and Combinatorics

Cited by 13 publications

References 26 publications

Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs

Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs

Classes of normally and nearly normally torsion-free monomial ideals

Waldschmidt Constants for Stanley–Reisner Ideals of a Class of Graphs

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