Related papers: An application of type sequences to the blowing-up

Let $(R, {\mathfrak m})$ be a Noetherian local ring and let $I$ be an ${\mathfrak m}$-primary ideal. In this paper we give sharp bounds on the multiplicity of the special fiber ring ${\mathcal F}$ of $I$ in terms of other well-known…

We extend the notion of type sequence to rings that are not necessarily residually rational. Using this invariant we characterize different types of rings as almost Gorenstein rings and rings of maximal length.

Let R be a one-dimensional local Noetherian domain, which is supposed analytically irreducible and residually rational, and let I be a proper ideal of R. Our purpose is to study the two numbers l(I^*/R)-l(R/I) and rl(R/I)-l(I^*/R) (l…

In this article we investigate when a complete ideal in a two-dimensional regular local ring is a multiplier ideal of some ideal with an integral multiplying parameter. In particular, we show that this question is closely connected to the…

We establish a new fundamental class of varieties in nonnoetherian algebraic geometry related to the central geometry of dimer algebras. Specifically, given an affine algebraic variety $X$ and a finite collection of non-intersecting…

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ a finitely generated $R$-module. We study the relations of the index of reducibility and the irreducible multiplicity of an $\mathfrak{m}$-primary ideal of $R$ and these of…

Let $R$ be a $d$-dimensional Noetherian local ring with maximal ideal $m_R$. In this article, we give a generalization of the multiplicity $e(I)$ of an $m_R$-primary ideal $I$ of $R$ to a multiplicity $e(\mathcal I)$ of a graded family of…

We extend the classical formula of Porteous for blowing-up Chern classes to the case of blow-ups of possibly singular varieties along regularly embedded centers. The proof of this generalization is perhaps conceptually simpler than the…

We initiate the study of the resolution of singularities properties of Nash blowups over fields of prime characteristic. We prove that the iteration of normalized Nash blowups desingularizes normal toric surfaces. We also introduce a prime…

We study the blow-ups X of P3 along a proj. normal curve C. We look for very ample divisor classes on X of low degree, and we study the ideal of the embedding of X. Some result is generalized to higher dimensions.

Let I be a finitely supported complete m-primary ideal of a regular local ring (R, m). We consider singularities of the normalization of the blow-up Proj R[It] of I. A theorem of Lipman implies that the ideal I has a unique factorization as…

By Perelman's $\mathcal L$-geodesic theory, we study the blow-down solutions on a noncompact $\kappa$-noncollapsed steady gradient Ricci soliton $(M^n, g)$ $(n\ge 4)$ with nonnegative curvature operator and positive Ricci curvature away…

Let V be an irreducible affine algebraic variety over a field k of characteristic zero, and let (f_0,...,f_m) be a sequence of elements of the coordinate ring. There is probably no elementary condition on the f_i and their derivatives which…

We study different notions of blow-up of a scheme X along a subscheme Y, depending on the datum of an embedding of X into an ambient scheme. The two extremes in this theory are the ordinary blow-up, corresponding to the identity, and the…

Many examples of nonpositively curved closed manifolds arise as blow-ups of projective hyperplane arrangements. If the hyperplane arrangement is associated to a finite reflection group W, and the blow-up locus is W-invariant, then the…

The paper is devoted to the analysis of the blow-ups of derivatives, gradient catastrophes and dynamics of mappings of $\mathbb{R}^n \to \mathbb{R}^n$ associated with the $n$-dimensional homogeneous Euler equation. Several characteristic…

Blow-ups of derivatives and gradient catastrophes for the $n$-dimensional homogeneous Euler equation are discussed. It is shown that, in the case of generic initial data, the blow-ups exhibit a fine structure in accordance of the admissible…

We associate a combinatorial object to sequences of point blow-ups over perfect fields, the weighted directed graph, and another one to the composition of all blow-ups, which we call associated sequential morphisms, the $d-$ary intersection…

We prove that two arbitrary ideals $I \subset J$ in an equidimensional and universally catenary Noetherian local ring have the same integral closure if and only if they have the same multiplicity sequence. We also obtain a Principle of…

We consider the Gelfand problem on a planar domain. Under some conditions on the potential, we provide the first examples of multiplicity for blowing-up solutions at a given point in the domain. The argument is based on a refined…