a presentation of M , where F is a free module of arbitrary rank. Let U denote the matrix (with n rows but possibly infinitely many columns) representing φ with respect to some fixed bases for F and An. For t ≥ 1 we let It(φ) denote the ideal of A generated by all t-sized minors U . (This ideal is independent of the choice of bases.) By convention, It(φ) = R for t ≤ 0 and It(φ) = 0 if t exceeds...

We establish a relation between topological and quantum entanglement for a multi-qubit state by considering the unitary representations of the Artin braid group. We construct topological operators that can entangle multi-qubit state. In particular we construct operators that create quantum entanglement for multi-qubit states based on the Segre ideal of complex multi-projective space. We also in...

1 In this paper we consider reduced homogeneous ideals J ⊂ S of a polynomial ring S, having a 2-linear resolution. 1. We study systems of generators of J ⊂ S. 2. We compute the arithmetical rank for a large class of projective curves having a 2-linear resolution. 3. We show that the fiber cone projF(IL) of a lattice ideal IL of codimension two is a set theoretical complete intersection.

Let $$\mathcal {D}$$ be a weighted oriented graph and let $$I(\mathcal {D})$$ its edge ideal in polynomial ring R. We give the formula of Castelnuovo–Mumford regularity $$R/I(\mathcal when is path or cycle such that edges are one direction. Additionally, we compute projective dimension for this class graphs.

Let G be a chordal graph and I(G) its edge ideal. Let β(I(G)) = (β0, β1, . . . , βp) denote the Betti sequence of I(G), where βi stands for the ith total Betti number of I(G) and where p is the projective dimension of I(G). It will be shown that there exists a simplicial complex ∆ of dimension p whose f -vector f(∆) = (f0, f1, . . . , fp) coincides with β(I(G)).

We show, under suitable hypothesis which are sharp in a certain sense, that the core of an m-primary ideal in a regular local ring of dimension d is equal to the adjoint (or multiplier) ideal of its d-th power. This generalizes the fundamental formula for the core of an integrally closed ideal in a two dimensional regular local ring due to Huneke and Swanson. We also find a generalization of th...

For an integer n ≥ 3, the clutter ∆n := { {1, 2}, {1, 3}, . . . , {1, n}, {2, 3, . . . , n} } is called a delta of dimension n, whose members are the lines of a degenerate projective plane. In his seminal paper on non-ideal clutters, Alfred Lehman manifested the role of the deltas as a distinct class of minimally non-ideal clutters [DIMACS, 1990]. A clutter is delta free if it has no delta mino...

The Chow variety, introduced in 1937 by Chow and van der Waerden [4], parameterizes algebraic cycles of any fixed dimension and degree in a projective space, each given by its Chow form. The case of curves in IP goes back to an 1848 paper by Cayley [3]. A fundamental problem, addressed by Green and Morrison [8] as well as Gel’fand, Kapranov and Zelevinsky [6, §4.3], is to describe the equations...

In this paper we give bounds on the Castelnuovo-Mumford regularity of products of ideals and ideal sheaves. In particular, we show that the regularity of a product of ideals is bounded by the sum of the regularities of its factors if the corresponding schemes intersect in a finite set of points. We also show how approximations of sheaves can be used to bound the regularity of an arrangement of ...

Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion of his 65th birthday. Abstract We study the fibers of projective morphisms and rational maps. We characterize the analytic spread of a homogeneous ideal through properties of its syzygy matrix. Powers of linearly presented ideals need not be linearly presented, but we identify a weaker li...