Related papers: Multiplicities and log canonical threshold

Let $K$ be an algebraically closed field. There has been much interest in characterizing multiple structures in $\P^n_K$ defined on a linear subspace of small codimension under additional assumptions (e.g. Cohen-Macaulay). We show that no…

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d \geq 2$ with infinite residue field and let $I$ be an $\mathfrak{m}$-primary ideal. For $0 \leq i \leq d$ let $I_i$ be the $i^{th}$-coefficient ideal of $I$. Also let…

Let $\mathscr{I}$ be an ideal sheaf on $P^n$. In the first part of this paper, we bound the asymptotic regularity of powers of $\mathscr{I}$ as $ps-3\leq \reg \mathscr{I}^p\leq ps+e$, where $e$ is a constant and $s$ is the $s$-invariant of…

We prove that the local accumulation complexity of the set of log canonical volumes in dimension $\geq 2$ can be infinite.

Let $(A,\mathfrak{m})$ be an analytically un-ramified Noetherian local ring of dimension $d \geq 1$, $I$ a regular $\mathfrak{m}$-primary ideal of $A$ and let $\overline{I}$ be integral closure ideal of $I$. If $A$ is of characteristic $p >…

Let $X$ be a smooth hypersurface of degree $n\geq 3$ in $\mathbb{P}^n$. We prove that the log canonical threshold of $H\in|-K_X|$ is at least $\frac{n-1}{n}$. Under the assumption of the Log minimal model program, we also prove that a…

Let $R$ be a commutative Noetherian ring with non-zero identity and $\fa$ an ideal of $R$. Let $M$ be a finite $R$--module of of finite projective dimension and $N$ an arbitrary finite $R$--module. We characterize the membership of the…

The long standing Lech's conjecture in commutative algebra states that for a flat local extension $(R,\mathfrak{m})\to (S,\mathfrak{n})$ of Noetherian local rings, we have an inequality on the Hilbert--Samuel multiplicities: $e(R)\leq…

Let $K$ be a field and let $R = K[X_1, \ldots, X_m]$ with $m \geq 2$. Give $R$ the standard grading. Let $I$ be a homogeneous ideal of height $g$. Assume $1 \leq g \leq m -1$. Suppose $H^i_I(R) \neq 0$ for some $i \geq 0$. We show (1)…

Let $(R,\mathfrak{m})$ be a Noetherian local ring, and let $M$ be a finitely generated $R$-module of dimension $d$. We prove that the set $\left\{\frac{l(M/IM)}{e(I, M)} \right\}_{\sqrt{I}=\mathfrak{m}}$ is bounded below by…

Let $R$ be a $d$-dimensional standard graded ring over an Artin local ring. Let $M$ be the unique maximal homogeneous ideal of $R.$ Let $h^i(R)_n$ denote the length of $H^i_M(R)_n$, i.e. the nth graded component of the ith local cohomology…

Let $R$ be a commutative noetherian ring, $I,J$ be two ideals of $R$, $M$ be an $R$-module, and $\mathcal{S}$ be a Serre class of $R$-modules. A positive answer to the Huneke$^,$s conjecture is given for a noetherian ring $R$ and minimax…

In characteristic zero, we construct relative principalization of ideals for logarithmically regular morphisms of logarithmic schemes, and use it to construct logarithmically regular desingularization of morphisms. These constructions are…

We study the regularity of symbolic powers of square-free monomial ideals. We prove that if $I = I_\Delta$ is the Stanley-Reisner ideal of a simplicial complex $\Delta$, then $\reg(I^{(n)}) \leqslant \delta(n-1) +b$ for all $n\geqslant 1$,…

Let i be a homomorphism of the multiplicative group into a connected reductive algebraic group over C. Let G^i be the centralizer of the image i. Let LG be the Lie algebra of G and let L_nG (n integer) be the summands in the direct sum…

For each irreducible finite dimensional representation of the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$ of $2\times 2$ traceless matrices, an explicit uniform upper bound is given for the multiplicities in the cocharacter sequence of the…

For any ideal $I$ in a Noetherian local ring or any graded ideal $I$ in a standard graded $K$-algebra over a field $K$, we introduce the socle module $\mathrm{Soc}(I)$, whose graded components give us the socle of the powers of $I$. It is…

We prove that the arithmetic degree of a graded or local ring is bounded above by the arithmetic degree of any of its associated graded rings with respect to ideals $I$ in $A$. In particular, if $Spec (A)$ is equidimensional and has an…

Let R denote a commutative Noetherian ring and let I be an ideal of R such that H_i^I(R) = 0, for all integers i greater than or equal to 2. In this paper we shall prove some results concerning the homological properties of I.

Let $(R, \mathfrak m)$ be a one dimensional local Cohen-Macaulay ring. An $\mathfrak m$-primary ideal $I$ of $R$ is Elias if the types of $I$ and of $R/I$ are equal. Canonical and principal ideals are Elias, and Elias ideals are closed…