Regularity and projective dimension of edge ideals

In this paper, we study the regularity and projective dimension of edge ideals. We provide two upper bounds for the regularity of edge ideals of vertex decomposable graphs in terms of the induced matching number and the number of cycles. Also, we generalize one of the upper bounds given by Dao and Schweig for the projective dimension of hypergraphs.     

Regularity of lex-segment ideals: Some closed formulas and applications

A closed formula for computing the regularity of the lex-segment ideal in terms of the Hilbert function is given. This regularity bounds the one of any ideal with the same Hilbert function. As a consequence, we give explicit expressions to bound the regularity of a projective scheme in terms of the coefficients of the Hilbert polynomial.

We also characterize, in terms of their coefficients, which polynomials are Hilbert polynomials of some projective scheme.

Finally, we provide some applications to estimates for the maximal degree of generators of Gröbner bases in terms of the degrees of defining equations.

     

Regularity, partial elimination ideals and the canonical bundle

We present partial elimination ideals, which set-theoretically cut out the multiple point loci of a generic projection of a projective variety, as a way to bound the regularity of a variety in projective space. To do this, we utilize a combination of initial ideal methods and geometric methods. We first define partial elimination ideals and establish through initial ideal methods the way in which, for a given ideal, the regularity of the partial elimination ideals bounds the regularity of the given ideal. Then we explore the partial elimination ideals as a way to compute the canonical bundle of the generic projection of a variety and the canonical bundles of the multiple point loci of the projection, and we use Kodaira Vanishing to bound the regularity of the partial elimination ideals.

     

On 3-syzygy and unexpected plane curves

Geometriae Dedicata - We prove a new inequality relating volume to length of closed geodesics on area minimizers for generic metrics on the complex projective plane. We exploit recent regularity...     

A generalization of Castelnuovo regularity to Grassmann varieties

Using the Beilinson–Kapranov spectral sequence, we define Castelnuovo regularity for a coherent sheaf on a Grassmann variety. We show that many formal properties of regularity over projective spaces continue to hold in this generality and derive concrete bounds on the degrees of defining equations for subvarieties of the Grassmannian. Received: 23 November 1999 / Revised version: 12 March 2000     

On the Resolution of Path Ideals of Cycles

We give a formula to compute all the top degree graded Betti numbers of the path ideal of a cycle. Also we will find a criterion to determine when Betti numbers of this ideal are nonzero and give a formula to compute its projective dimension and regularity.     

Classification and syzygies of smooth projective varieties with 2-regular structure sheaf

The geometric and algebraic properties of smooth projective varieties with 1-regular structure sheaf are well understood, and the complete classification of these varieties is a classical result. The aim of this paper is to study the next case: smooth projective varieties with 2-regular structure sheaf. First, we give a classification of such varieties using adjunction mappings. Next, under suitable conditions, we study the syzygies of section rings of those varieties to understand the structure of the Betti tables, and show a sharp bound for Castelnuovo–Mumford regularity.     

Asymptotic syzygies of algebraic varieties

We study the asymptotic behavior of the syzygies of a smooth projective variety as the positivity of the embedding line bundle grows. The main result asserts that the syzygy modules are non-zero in almost all degrees allowed by Castelnuovo?CMumford regularity. We also give an effective statement for Veronese varieties that we conjecture to be optimal.     

On the Regularity of Codimension Two Surfaces and Threefolds with Singular Loci

ABSTRACT

For projective codimension two surfaces and threefolds whose singular locus is one dimensional, we get the sharp Castelnuovo–Mumford regularity bound in terms of degrees of defining equations and give the classification of nearly extremal cases. This is a generalization of the result of Bertram et al.     

Sharp bounds on Castelnuovo-Mumford regularity

The Castelnuovo-Mumford regularity is one of the most important invariants in studying the minimal free resolution of the defining ideals of the projective varieties. There are some bounds on the Castelnuovo-Mumford regularity of the projective variety in terms of the other basic measures such as dimension, codimension and degree. In this paper we consider an upper bound on the regularity of a nondegenerate projective variety , , provided is -Buchsbaum for , and investigate the projective variety with its Castelnuovo-Mumford regularity having such an upper bound.

     

On strongly (<Emphasis Type="Italic">P</Emphasis>)-cyclic acts

By a regular act we mean an act such that all its cyclic subacts are projective. In this paper we introduce strong (P)-cyclic property of acts over monoids which is an extension of regularity and give a classification of monoids by this property of their right (Rees factor) acts.     

A bound on the Castelnuovo-Mumford regularity for curves

Recall that a projective curve in with ideal sheaf is said to be n-regular if for every integer and that in this case, it is cut out scheme-theoretically by equations of degree at most n. The purpose here is to show that an irreducible, reduced, projective curve of degree d and large arithmetic genus satisfies a smaller regularity bound than the optimal one . For example, if then a curve is -regular unless it is embedded by a complete linear system of degree . Received: 29 May 2000 / Published online: 24 September 2001     

On the regularity over positively graded algebras

We study the relationship between the Tor-regularity and the local-regularity over a positively graded algebra defined over a field which coincide if the algebra is a standard graded polynomial ring. In this case both are characterizations of the so-called Castelnuovo–Mumford regularity. Moreover, we can characterize a standard graded polynomial ring as a K-algebra with extremal properties with respect to the Tor- and the local-regularity. For modules of finite projective dimension we get a nice formula relating the two regularity notions. Interesting examples are given to help to understand the relationship between the Tor- and the local-regularity in general.     

Regularity,depth and arithmetic rank of bipartite edge ideals

We study minimal free resolutions of edge ideals of bipartite graphs. We associate a directed graph to a bipartite graph whose edge ideal is unmixed, and give expressions for the regularity and the depth of the edge ideal in terms of invariants of the directed graph. For some classes of unmixed edge ideals, we show that the arithmetic rank of the ideal equals projective dimension of its quotient.     

Algebraic Properties of Edge Ideals of Some Vertex-Weighted Oriented Cyclic Graphs

We provide some exact formulas for the projective dimension and regularity of edge ideals associated to some vertex-weighted oriented cyclic graphs with a common vertex or edge.These formulas are functions in the weight of the vertices,and the numbers of edges and cycles.Some examples show that these formulas are related to direction selection and the assumption that w(x)≥2 for any vertex x cannot be dropped.     

Homological properties of Orlik–Solomon algebras

The Orlik–Solomon algebra of a matroid can be considered as a quotient ring over the exterior algebra E. At first, we study homological properties of E-modules as e.g., complexity, depth and regularity. In particular, we consider modules with linear injective resolutions. We apply our results to Orlik–Solomon algebras of matroids and give formulas for the complexity, depth and regularity of such rings in terms of invariants of the matroid. Moreover, we characterize those matroids whose Orlik–Solomon ideal has a linear projective resolution and compute in these cases the Betti numbers of the ideal.     

Some Koszul rings from geometry

We give examples of Koszul rings that arise naturally in algebraic geometry. In the first part, we prove a general result on Koszul property associated to an ample line bundle on a projective variety. Specifically, we show how Koszul property of multiples of a base point free ample line bundle depends on its Castelnuovo–Mumford regularity. In the second part, we give examples of Koszul rings that come from adjoint line bundles on minimal irregular surfaces of general type.     

Second-order conditions for nonsmooth multiobjective optimization problems with inclusion constraints

This paper investigates second-order optimality conditions for general multiobjective optimization problems with constraint set-valued mappings and an arbitrary constraint set in Banach spaces. Without differentiability nor convexity on the data and with a metric regularity assumption the second-order necessary conditions for weakly efficient solutions are given in the primal form. Under some additional assumptions and with the help of Robinson -Ursescu open mapping theorem we obtain dual second-order necessary optimality conditions in terms of Lagrange-Kuhn-Tucker multipliers. Also, the second-order sufficient conditions are established whenever the decision space is finite dimensional. To this aim, we use the second-order projective derivatives associated to the second-order projective tangent sets to the graphs introduced by Penot. From the results obtained in this paper, we deduce and extend, in the special case some known results in scalar optimization and improve substantially the few results known in vector case.     

On Castelnuovo-Mumford regularity of projective curves

We give an effective method to compute the regularity of a saturated ideal defining a projective curve that also determines in which step of a minimal graded free resolution of the regularity is attained.