The Castelnuovo-Mumford regularity of a graded ring is an important invariant in computational commutative algebra, and there is increasing interest in multigraded generalizations. We study connections between two recent definitions of multigraded regularity with a view towards a better understanding of the multigraded Hilbert function of fat point schemes in P n1 × · · · × P k . Introduction L...

let $l$ be a lattice in $zz^n$ of dimension $m$. we prove that there exist integer constants $d$ and $m$ which are basis-independent such that the total degree of any graver element of $l$ is not greater than $m(n-m+1)md$. the case $m=1$ occurs precisely when $l$ is saturated, and in this case the bound is a reformulation of a well-known bound given by several authors. as a corollary, we show t...

We show that the Castelnuovo–Mumford regularity of the binomial edge ideal of a graph is bounded below by the length of its longest induced path and bounded above by the number of its vertices.

The author continues the study of Frobenius amplitude, introduced in an earlier paper, and compares this to various other positivity notions. These are applied to deduce vanishing theorems. A refinement of Castelnuovo-Mumford regularity is also given.

Castelnuovo–Mumford regularity is one of the most important invariants in commutative algebra. It has been investigated by various authors from different point of view. One of the interesting properties of regularity is its asymptotic behavior. In this talk first we will see some well-known facts of regularity and then I will present some known results of asymptotic behavior of quadratic square...

We prove that for every integer \(k\ge 1\), there exists a connected graph \(H_k\) such \(v(H_k)={\text {reg}}(H_k)+k\), where v(G) and \({\text {reg}}(G)\) denote the v-number (Castelnuovo–Mumford) regularity of G respectively.

Let I be a homogeneous ideal of the polynomial ring K[x0, . . . , xn], where K is an arbitrary field. Avoiding the construction of a minimal graded free resolution of I, we provide effective methods for computing the Castelnuovo-Mumford regularity of I that also compute other cohomological invariants of K[x0, . . . , xn]/I. We then apply our methods to the defining ideal I(V) of a projective mo...

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...