Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · alg-geom/9602021

Bounds on cohomology and Castelnuovo-Mumford regularity

Chikashi Miyazaki·Wolfgang Vogel
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Castelnuovo-Mumford regularity reg(X) of a projective scheme X was introduced by Mumford by generalizing ideas of Castelnuovo. The interest in this concept stems partly from the fact that X is m-regular if and only if for every p \geq 0 the minimal generators of the p-th syzygy module of the defining ideal I of X occur in degree \leq m + p. There are some bounds in the case that X is a locally Cohen-Macaulay scheme. The aim of this paper is to extend and improve these results for so-called (k,r)-Buchsbaum schemes. In order to prove our theorems, we need to apply a spectral sequence. We conclude by describing two sharp examples and open problems.

Explore related subjects

alg-geommath.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Chikashi Miyazaki, Wolfgang Vogel. 1996-02-28. Bounds on cohomology and Castelnuovo-Mumford regularity. https://arxiv.org/abs/alg-geom/9602021

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2019

Classification of Varieties with Canonical Curve Section via Gaussian maps on Canonical Curves

Let $C \subset P^{g-1}$ be a smooth canonical curve of genus $g \geq 3$. The purpose of this article is to further develop a method to classify varieties having $C$ as their curve section, using Gaussian map computations. In a previous article a careful analysis of the degeneration to the cone over the hyperplane section was made for _prime_ Fano threefolds, that is Fano threefolds whose Picard group is generated by the hyperplane bundle. In this article we extend this method and classify Fano threefolds of higher index (which still have Picard number one). We are also able to classify Mukai varieties, i.e. varieties of dimension four or more with canonical curve sections.

alg-geom↗
arXiv2017

Boundedness and $K^2$ for log surfaces

Let $ε, C$ be two positive real numbers, and $\mathcal C \subset \mathbb R$ be a DCC (descending chain condition) set. Let $(X, B = \sum b_j B_j)$ denote a projective surface with an $\mathbb R$-divisor. Then (1) The class $\{X\}$ of surfaces for which there exists a divisor $B$ such that $(X,B)$ is $ε$-log terminal and $-(K_X + B)$ is nef (excluding only those for which at the same time $K_X\equiv 0$, $B=0$, and $X$ has at worst Du Val singularities), is bounded. (2) The set $\{(K_X + B)^2\}$ of squares for the semi log canonical pairs $(X, B)$ with ample $K_X + B$ and $b_j \in \mathcal C$, is a DCC set. (3) The class $\{(X,B)\}$ of pairs such that $(X, B)$ is semi log canonical, $K_X + B$ is ample, $(K_X + B)^2 = C$ and $b_j \in \mathcal C$, is bounded.

alg-geom↗