Asymptotic linear bounds for the Castelnuovo-Mumford regularity

We prove asymptotic linear bounds for the Castelnuovo-Mumford regularity of certain filtrations of homogeneous ideals whose Rees algebras need not be Noetherian.

     

Castelnuovo-Mumford regularity by approximation

The Castelnuovo-Mumford regularity of a module gives a rough measure of its complexity. We bound the regularity of a module given a system of approximating modules whose regularities are known. Such approximations can arise naturally for modules constructed by inductive combinatorial means. We apply these methods to bound the regularity of ideals constructed as combinations of linear ideals and the module of derivations of a hyperplane arrangement as well as to give degree bounds for invariants of finite groups.     

Finiteness of Hilbert functions and bounds for Castelnuovo-Mumford regularity of initial ideals

Bounds for the Castelnuovo-Mumford regularity and Hilbert coefficients are given in terms of the arithmetic degree (if the ring is reduced) or in terms of the defining degrees. From this it follows that there exists only a finite number of Hilbert functions associated with reduced algebras over an algebraically closed field with a given arithmetic degree and dimension. A good bound is also given for the Castelnuovo-Mumford regularity of initial ideals which depends neither on term orders nor on the coordinates and holds for any field.

     

Asymptotic behaviour of Castelnuovo-Mumford regularity

Let be a polynomial ring over a field. For a graded -module generated in degree at most , the Castelnuovo-Mumford regularity of each of (i) its symmetric power, (ii) its torsion-free symmetric power and (iii) the integral closure of its torsion-free symmetric power is bounded above by a linear function in with leading coefficient at most . For a graded ideal of , the regularity of is given by a linear function of for all sufficiently large . The leading coefficient of this function is identified.

     

Castelnuovo-Mumford regularity and extended degree

Our main result shows that the Castelnuovo-Mumford regularity of the tangent cone of a local ring is effectively bounded by the dimension and any extended degree of . From this it follows that there are only a finite number of Hilbert-Samuel functions of local rings with given dimension and extended degree.

     

Castelnuovo-Mumford regularity for nonhyperelliptic curves

The purpose here is to show that an irreducible, reduced, projective, nonhyperelliptic curve of degree d and genus g is n-regular for if Received: 10 July 2003     

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.

     

Evaluations of initial ideals and Castelnuovo-Mumford regularity

This paper characterizes the Castelnuovo-Mumford regularity by evaluating the initial ideal with respect to the reverse lexicographic order.

     

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 Castelnuovo-Mumford regularity of connected curves

In this paper we prove that the regularity of a connected curve is bounded by its degree minus its codimension plus 1. We also investigate the structure of connected curves for which this bound is optimal. In particular, we construct connected curves of arbitrarily high degree in having maximal regularity, but no extremal secants. We also show that any connected curve in of degree at least 5 with maximal regularity and no linear components has an extremal secant.

     

Bounds on Castelnuovo-Mumford regularity for generalized Cohen-Macaulay graded rings

Supported by a grant from the Centre de Recerca Matematica, Institut d'Estudis Catalans (Spain)     

Castelnuovo-Mumford regularity for complexes and weakly Koszul modules

Let A be a noetherian AS-regular Koszul quiver algebra (if A is commutative, it is essentially a polynomial ring), and the category of finitely generated graded left A-modules. Following Jørgensen, we define the Castelnuovo-Mumford regularity of a complex in terms of the local cohomologies or the minimal projective resolution of M^•. Let A^! be the quadratic dual ring of A. For the Koszul duality functor , we have . Using these concepts, we interpret results of Martinez-Villa and Zacharia concerning weakly Koszul modules (also called componentwise linear modules) over A^!. As an application, refining a result of Herzog and Römer, we show that if J is a monomial ideal of an exterior algebra E=?〈y₁,…,y_d〉, d≥3, then the (d−2)nd syzygy of E/J is weakly Koszul.     

Castelnuovo-Mumford regularity of simplicial semigroup rings with isolated singularity

Let be the polynomial ring in variables over a field and its graded maximal ideal. Let be homogeneous polynomials of degree generating an -primary ideal, and let be arbitrary homogeneous polynomials of degree . In the present paper it will be proved that the Castelnuovo-Mumford regularity of the standard graded -algebra is at most . By virtue of this result, it follows that the regularity of a simplicial semigroup ring with isolated singularity is at most , where is the multiplicity of and is the codimension of .

     

Castelnuovo-Mumford regularity and projective dimension of a squarefree monomial ideal

Let S = K[x₁; x₂;...; x_n] be the polynomial ring in n variables over a field K; and let I be a squarefree monomial ideal minimally generated by the monomials u₁; u₂;...; u_m: Let w be the smallest number t with the property that for all integers 1 6 i₁ < i₂ <... < i _t 6 m such that \(lcm({u_{{i_1}}},{u_{{i_2}}},...,{u_{{i_t}}}) = lcm({u_1},{u_2},...,{u_m})\) We give an upper bound for Castelnuovo-Mumford regularity of I by the bigsize of I: As a corollary, the projective dimension of I is bounded by the number w.     

A sharp bound for the Castelnuovo-Mumford regularity of subspace arrangements

We show that the ideal of an arrangement of d linear subspaces of projective space is d-regular in the sense of Castelnuovo and Mumford, answering a question of B. Sturmfels. In particular, this implies that the ideal of an arrangement of d subspaces is generated in degrees less than or equal to d.     

A bound for the Castelnuovo-Mumford regularity of log canonical varieties

In this note, we give a bound for the Castelnuovo-Mumford regularity of a homogeneous ideal I in terms of the degrees of its generators. We assume that I defines a local complete intersection with log canonical singularities.     

Comparing Castelnuovo-Mumford regularity and extended degree: The borderline cases

Castelnuovo-Mumford regularity and any extended degree function can be thought of as complexity measures for the structure of finitely generated graded modules. A recent result of Doering, Gunston, and Vasconcelos shows that both can be compared in the case of a graded algebra. We extend this result to modules and analyze when the estimate is in fact an equality. A complete classification is obtained if we choose as extended degree the homological or the smallest extended degree. The corresponding algebras are characterized in three ways: by relations among the algebra generators, by using generic initial ideals, and by their Hilbert series.