Combinatorics of Toric Arrangements

In this paper we build an Orlik-Solomon model for the canonical gradation of the cohomology algebra with integer coefficients of the complement of a toric arrangement. We give some results on the uniqueness of the representation of arithmetic matroids, in order to discuss how the Orlik-Solomon model depends on the poset of layers. The analysis of discriminantal toric arrangements permits us to isolate certain conditions under which two toric arrangements have diffeomorphic complements. We also give combinatorial conditions determining whether the cohomology algebra is generated in degree one.Comment: 29 pages, 1 figur

The Orlik-Solomon model for hypersurface arrangements

We develop a model for the cohomology of the complement of a hypersurface arrangement inside a smooth projective complex variety. This generalizes the case of normal crossing divisors, discovered by P. Deligne in the context of the mixed Hodge theory of smooth complex varieties. Our model is a global version of the Orlik-Solomon algebra, which computes the cohomology of the complement of a union of hyperplanes in an affine space. The main tool is the complex of logarithmic forms along a hypersurface arrangement, and its weight filtration. Connections with wonderful compactifications and the configuration spaces of points on curves are also studied.Comment: 23 pages; presentation simplified, results unchange

Generalizing the Borel property

We introduce the notion of Q-Borel ideals: ideals which are closed under the Borel moves arising from a poset Q. We study decompositions and homological properties of these ideals, and offer evidence that they interpolate between Borel ideals and arbitrary monomial ideals.Comment: 19 pages, 1 figur

The graph of equivalence classes of zero-divisors of a poset

In this paper, we give the definition of the graph of equivalence classes of zero-divisors of a poset P. We prove that if [a] has maximal degree in V(γE(P)), then ann(a) is maximal in Anih(P). Also, we give some other properties of the graph γE(P). Moreover, we characterize the cut vertices of γE(P) and study the cliques of these graphs

Component graphs of vector spaces and zero-divisor graphs of ordered sets

In this paper, nonzero component graphs and nonzero component union graphs of finite dimensional vector space are studied using the zero-divisor graph of specially constructed 0-1-distributive lattice and the zero-divisor graph of rings. Further, we define an equivalence relation on nonzero component graphs and nonzero component union graphs to deduce that these graphs are the graph join of zero-divisor graphs of Boolean algebras and complete graphs. In the last section, we characterize the perfect and chordal nonzero component graphs and nonzero component union graphs.Comment: arXiv admin note: text overlap with arXiv:2205.0491

Pure simplicial complexes and well-covered graphs

A graph $G$ is called well-covered if all maximal independent sets of vertices have the same cardinality. A simplicial complex $\Delta$ is called pure if all of its facets have the same cardinality. Let $\mathcal G$ be the class of graphs with some disjoint maximal cliques covering all vertices. In this paper, we prove that for any simplicial complex or any graph, there is a corresponding graph in class $\mathcal G$ with the same well-coveredness property. Then some necessary and sufficient conditions are presented to recognize fast when a graph in the class $\cal G$ is well-covered or not. To do this characterization, we use an algebraic interpretation according to zero-divisor elements of the edge rings of graphs.Comment: 10 pages. arXiv admin note: substantial text overlap with arXiv:1009.524