Let m12, m13, . . . , mn−1,n be the slopes of the ( n 2 ) lines connecting n points in general position in the plane. The ideal In of all algebraic relations among the mij defines a configuration space called the slope variety of the complete graph. We prove that In is reduced and Cohen-Macaulay, give an explicit Gröbner basis for it, and compute its Hilbert series combinatorially. We proceed c...

We introduce and study the notions of conical and spherical graphs. We show that these mutually exclusive properties, which have a geometric interpretation, provide links between apparently unrelated classical concepts such as dominating sets, independent dominating sets, edge covers, and the homotopy type of an associated simplicial complex. In particular, we solve the problem of characterizin...

Let ∆ be a simplicial complex and I∆ its Stanley–Reisner ideal. It has been conjectured that, for each i and j, the graded Betti number βii+j(I∆) of I∆ is smaller than or equal to that of I∆c , where ∆ c is a combinatorial shifted complex of ∆. In the present paper the conjecture will be proved affirmatively. In particular the inequalities βii+j(I∆) ≤ βii+j(I∆lex) hold for all i and j, where ∆ ...

We study the relationship between the projective dimension of a squarefree monomial ideal and the domination parameters of the associated graph or clutter. In particular, we show that the projective dimensions of graphs with perfect dominating sets can be calculated combinatorially. We also generalize the wellknown graph domination parameter τ to clutters, obtaining bounds on the projective dim...

In this note we generalize and unify two results on connectivity of graphs: one by Balinsky and Barnette, one by Athanasiadis. This is done through a simple proof using commutative algebra tools. In particular we use bounds for the Castelnuovo–Mumford regularity of their Stanley–Reisner rings. As a result, if ∆ is a simplicial d-pseudomanifold and s is the largest integer such that ∆ has a miss...

For a graph of an n-cycle ∆ with Alexander dual ∆, we study the free resolution of a subideal G(n) of the Stanley-Reisner ideal I∆∗ . We prove that if G(n) is generated by 3 generic elements of I∆∗ , then the second syzygy module of G(n) is isomorphic to the second syzygy module of (x1, x2, . . . , xn). A result of Bruns shows that there is always a 3-generated ideal with this property. We show...

Matrix Schubert varieties are the closures of orbits B×B acting on all n×n matrices, where B is group invertible lower triangular matrices. Extending work Fulton, Knutson and Miller identified a Gröbner basis for prime ideals these varieties. They also showed that corresponding initial Stanley-Reisner shellable simplicial complexes, derived related primary decomposition in terms reduced pipe dr...

In this paper we study the Betti numbers of Stanley-Reisner ideals generated in degree 2. We show that the first six Betti numbers do not depend on the characteristic of the ground field. We also show that, if the number of variables n is at most 10, all Betti numbers are independent of the ground field. For n = 11, there exists precisely 4 examples in which the Betti numbers depend on the grou...