Ngo Viet Trung

Asymptotic Behaviour of the Castelnuovo-Mumford Regularity

In this paper the asymptotic behavior of the Castelnuovo

Standard graded vertex cover algebras, cycles and leaves

ndash;Mumford regularity of powers of a homogeneous ideal I is studied. It is shown that there is a linear bound for the regularity of the powers I whose slope is the maximum degree of a homogeneous generator of I, and that the regularity of I is a linear function for large n. Similar results hold for the integral closures of the powers of I. On the other hand we give examples of ideal for which the regularity of the saturated powers is asymptotically not a linear function, not even a linear function with periodic coefficients.

A sharp bound for the regularity index of fat points in general position

The aim of this paper is to characterize simplicial complexes which have standard graded vertex cover algebras. This property has several nice consequences for the squarefree monomial ideals defining these algebras. It turns out that such simplicial complexes are closely related to a range of hypergraphs which generalize bipartite graphs and trees. These relationships allow us to obtain very general results on standard graded vertex cover algebras which cover previous major results on Rees algebras of squarefree monomial ideals.

Absolutely superficial sequences

A bound is given for the regularity index of the coordinate ring of a set of fat points in general position in Pkn . The bound is attained by points on a rational normal curve.

On the Castelnuovo-Mumford regularity and the arithmetic degree of monomial ideals

Absolutely superficial sequences was introduced by P. Schenzel in order to study generalized Cohen-Macaulay (resp. Buchsbaum) modules. For an arbitrary local ring, they turned out to be d-sequences. This paper established properties of absolutely superficial sequences with respect to a module. It is shown that they are closely related to other sequences in the theory of generalized Cohen-Macaulay (resp. Buchsbaum) modules. In particular, there is a bounding function for the Hilbert-Samuel function of every parameter ideal such that this bounding function is attained if and only if the ideal is generated by an absolutely superficial sequence.

Castelnuovo-Mumford regularity and extended degree

Abstract In the first part of this paper we show that the Castelnuovo-Mumford regularity of a monomial ideal is bounded above by its arithmetic degree. The second part gives upper bounds for the Castelnuovo-Mumford regularity and the arithmetic degree of a monomial ideal in terms of the degrees of its generators. These bounds can be formulated for an arbitrary homogeneous ideal in terms of any Gröbner basis.

Positivity of mixed multiplicities

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.

On the core of ideals

Let R = ⊕(u,v)∈N2R(u,v) be a standard bigraded algebra over an artinian local ring K = R(0,0). Standard means R is generated over K by a finite number of elements of degree (1, 0) and (0, 1). Since the length (R(u,v)) of R(u,v) is finite, we can consider (R(u,v)) as a function in two variables u and v. This function was first studied by van der Waerden [W] and Bhattacharya [B] who proved that there is a polynomial PR(u, v) of (total) degree ≤ dimR − 2 such that (R(u,v)) = PR(u, v) for u and v large enough. Katz, Mandal andVerma [KMV] found out that the degree of PR(u, v) is equal to rdimR− 2, where rdimR is the relevant dimension of R defined as follows. Let R++ denote the ideal generated by the homogeneous elements of degree (u, v) with u ≥ 1, v ≥ 1. Let ProjR be the set of all homogeneous prime ideals ℘ ⊇ R++ of R. Then rdimR := max{dimR/℘| ℘ ∈ ProjR} if ProjR = ∅ and rdimR can be any negative integer if ProjR = ∅. If we write

On the multiplicity of blow-up rings of ideals generated by d-sequences

of I, denoted bycore(I), is defined to be the intersection of all reductions of I.The core of ideals was first studied by Rees and Sally [RS], partly due to itsconnection to the theorem of Brianc¸on and Skoda. Later, Huneke and Swanson[HuS] determined the core of integrally closed ideals in two-dimensional regularlocal rings and showed a close relationship to Lipman’s adjoint ideal. Recently,Corso, Polini and Ulrich [CPU1,2] gave explicit descriptions for the core of certainideals in Cohen-Macaulay local rings, extending the result of [HuS]. In these twopapers, several questions and conjectures were raised which provided motivationfor our work. More recently, Hyry and Smith [HyS] have shown that the core andits properties are closely related to a conjecture of Kawamata on the existence ofsections for numerically effective line bundles which are adjoint to an ample linebundle over a complex smooth algebraic variety, and they generalize the result in[HuS] to arbitrary dimension and more general rings. Nonetheless, there are manyunanswered questions on the nature of the core. One reason is that it is difficult todetermine the core and there are relatively few computed examples.Our focus in this paper is in effective computation of the core with an eye topartially answering some questions raised in [CPU1,2]. A first approach to under-standing the core was given by Rees and Sally. For an ideal Iin a local Noetherianring (R,m)having analytic spread l, one can take lgeneric generators ofIin aringofthe form R[U

Depth and regularity of powers of sums of ideals

We consider the Rees ring, the extended Rees ring and the associated graded ring of an ideal which is generated by a homogenous d-sequence. We introduce on these rings certain fine filtrations which allow us to compute their multiplicity with respect to their unique graded maximal ideal.