On the Chern number of II-admissible filtrations of ideals

Let $I$ be an \m-primary ideal of a Noetherian local ring (R, \m) of positive dimension. The coefficient $e_1(\mathcal I)$ of the Hilbert polynomial of an $I$-admissible filtration $\mathcal I$ is called the Chern number of $\mathcal I$. A formula for the Chern number has been derived involving Euler characteristic of subcomplexes of a Koszul complex. Specific formulas for the Chern number have been given in local rings of dimension at most two. These have been used to provide new and unified proofs of several results about $e_1(\mathcal I)$

Negativity of the Chern number of parameter ideals

In this paper we survey the results on the negativity of the Chern number of parameter ideals

A study of vv-number for some monomial ideals

In this paper, we give formulas for $v$-number of edge ideals of some graphs like path, cycle, 1-clique sum of a path and a cycle, 1-clique sum of two cycles and join of two graphs. For an $\mathfrak{m}$-primary monomial ideal $I\subset S=K[x_1,\ldots,x_t]$, we provide an explicit expression of $v$-number of $I$, denoted by $v(I)$, and give an upper bound of $v(I)$ in terms of the degree of its generators. We show that for a monomial ideal $I$, $v(I^{n+1})$ is bounded above by a linear polynomial for large $n$ and for certain classes of monomial ideals, the upper bound is achieved for all $n\geq 1$. For $\mathfrak m$-primary monomial ideal $I$ we prove that $v(I)\leq$ reg$(S/I)$ and their difference can be arbitrarily large.Comment: 15 page

Bounds for the reduction number of primary ideal in dimension three

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d\geq 3$ and $I$ an $\mathfrak{m}$-primary ideal of $R$. Let $r_J(I)$ be the reduction number of $I$ with respect to a minimal reduction $J$ of $I$. Suppose depth $G(I)\geq d-3$. We prove that $r_J(I)\leq e_1(I)-e_0(I)+\lambda(R/I)+1+(e_2(I)-1)e_2(I)-e_3(I)$, where $e_i(I)$ are Hilbert coefficients. Suppose $d=3$ and depth $G(I^t)>0$ for some $t\geq 1$. Then we prove that $r_J(I)\leq e_1(I)-e_0(I)+\lambda(R/I)+t$.Comment: To appear in Proc. Amer. Math. So

Bounds on the Castelnuovo-Mumford Regularity in dimension two

Consider a Cohen-Macaulay local ring $(R,\mathfrak m)$ with dimension $d\geq 2$, and let $I \subseteq R$ be an $\mathfrak m$-primary ideal. Denote $r_{J}(I)$ as the reduction number of $I$ with respect to a minimal reduction $J$ of $I$, and $\rho(I)$ as the stability index of the Ratliff-Rush filtration with respect to $I$. In this paper, we derive a bound on $\rho(I)$ in terms of the Hilbert coefficients and $r_{J}(I)$. In the case of two-dimensional Cohen-Macaulay local rings, the established bound on $\rho(I)$ consequently leads to a bound on the Castelnuovo-Mumford regularity of the associated graded ring of $I$.Comment: 16 Page