Irrational asymptotic behaviour of Castelnuovo-Mumford regularity

Multigraded Castelnuovo-mumford Regularity

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 m...

On the Asymptotic Linearity of Castelnuovo-mumford Regularity

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 an approximation by 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 m...

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 of Products of Ideals

This is not the case in general. There are examples already with M = I such that reg(I) > 2 reg(I), see Sturmfels [15] and Terai [16]. On the other hand, Chandler [5] and Geramita, Gimigliano and Pitteloud [11] have shown that reg(I) ≤ k reg(I) holds for ideals with dimR/I ≤ 1. In general one has that reg(I) is asymptotically a linear function of k, see [14, 8]. If one takes I = m and M any gra...