Showing papers on "Castelnuovo–Mumford regularity published in 2003"

Castelnuovo-Mumford regularity of products of ideals

Abstract: The Castelnuovo-Mumford regularity reg$(M)$ is one of the most important invariants of a finitely generated graded module $M$ over a polynomial ring $R$. For instance, it measures the amount of computational resources that working with $M$ requires. In general one knows that the regularity of a module can be doubly exponential in the degrees of the minimal generators and in the number of the variables. On the other hand, in many situations one has or one conjectures a much better behavior. One may ask, for instance, whether the Castelnuovo-Mumford regularity reg($IM$) of the product of an ideal $I$ with a module $M$ is bounded by the sum reg($I$) + reg($M$). In general this is not the case. But we show that it is indeed the case if either dim $R/I\leq 1$ or $I$ is generic (in a very precise sense). Further we show that products of ideals of linear forms have always a linear resolution and that the same is true for products of determinantal ideals of a generic Hankel matrix.

Multigraded Castelnuovo-Mumford Regularity

Abstract: We develop a multigraded variant of Castelnuovo-Mumford regularity. Motivated by toric geometry, we work with modules over a polynomial ring graded by a finitely generated abelian group. As in the standard graded case, our definition of multigraded regularity involves the vanishing of graded components of local cohomology. We establish the key properties of regularity: its connection with the minimal generators of a module and its behavior in exact sequences. For an ideal sheaf on a simplicial toric variety X, we prove that its multigraded regularity bounds the equations that cut out the associated subvariety. We also provide a criterion for testing if an ample line bundle on X gives a projectively normal embedding.

Castelnuovo-Mumford regularity and extended degree

Abstract: Our main result shows that the Castelnuovo-Mumford regularity of the tangent cone of a local ring A is effectively bounded by the dimension and any extended degree of A. 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 of simplicial semigroup rings with isolated singularity

Abstract: Let S = K[x 1 ,...,x n ] be the polynomial ring in n ≥ 2 variables over a field K and m its graded maximal ideal. Let f 1 f m ∈ S be homogeneous polynomials of degree d - 1 > 2 generating an m-primary ideal, and let g 1 ,...,g r E S be arbitrary homogeneous polynomials of degree d. In the present paper it will be proved that the Castelnuovo-Mumford regularity of the standard graded K-algebra A = K[{f i x j } i=1,...,m, g1,...,gr ] j=1,...,n is at most (d - 2)(n - 1). By virtue of this result, it follows that the regularity of a simplicial semigroup ring K[C] with isolated singularity is at most e(K[C]) - codim(K[C]), where e(K[C]) is the multiplicity of K[C] and codim(K[C]) is the codimension of K[C].

On the Castelnuovo-Mumford regularity of connected curves

Abstract: In this paper we prove the Eisenbud-Goto conjecture for connected curves. 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 projective 4-space having maximal regularity, but no extremal secants. We also show that any connected curve in projective 3-space of degree at least 5 that has no linear components and has maximal regularity has an extremal secant.

Castelnuovo-Mumford regularity and degrees of generators of graded submodules

Abstract: We extend the regularity criterion of Bayer-Stillman for a graded ideal $\mathfrak {a}$ of a polynomial ring $K[\underline {\bf x}] := K [\underline {\bf x}_0, \dots , {\bf x}_r]$ over an infinite field $K$ to the situation of a graded submodule $M$ of a finitely generated graded module $U$ over a Noetherian homogeneous ring $R = \oplus_{n \geq 0}R_n$, whose base ring $R_0$ has infinite residue fields. If $R_0$ is Artinian, we construct a polynomial $\widetilde{P} \in {\mathbb Q}[{\bf x}]$, depending only on the Hilbert polynomial of $U$, such that $\operatorname{reg}(M) \leq \widetilde{P} ( \max \{ d(M), \operatorname{reg}(U) + 1 \} ) $, where $d(M)$ is the generating degree of $M$. This extends the regularity bound of Bayer-Mumford for a graded ideal $\mathfrak {a} \subseteq K[\underline {\bf x}]$ over a field $K$ to the pair $M \subseteq U$.

Bounds for Castelnuovo-Mumford Regularity in Terms of Degrees of Defining Equations

Abstract: We present here some of our (often shared) work on Castelnuovo-Mumford regularity, give few applications to Grobner basis theory and show some examples.

Characteristic-free bounds for the Castelnuovo-Mumford regularity

Abstract: We study bounds for the Castelnuovo-Mumford regularity of homogeneous ideals in a polynomial ring in terms of the number of variables and the degree of the generators. In particular our aim is to give a positive answer to a question posed by Bayer and Mumford, by showing that the known upper bound in characteristic zero holds true also in positive characteristic. We first analyze Giusti's proof, which provides the result in characteristic 0, giving some insight on the combinatorial properties needed in that context. For the general case we provide a new argument which employs Bayer and Stillman criterion for detecting regularity.

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

Abstract: 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, Vasconcelos shows that both can be compared in 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.

Products of Ideals with Linear Resolution

Abstract: We will discuss the behavior of the Castelnuovo-Mumford regularity under certain operations on ideals and modules, like products or powers. In particular, we will see that reg(JM) can be larger than reg(M) + reg(I) even when I is an ideal of linear forms and M is a module with a linear resolution. On the other hand, we have shown that any product of ideals of linear forms has a linear resolution. We will also discuss the case of polymatroidal ideals.