Regularity of smooth curves in biprojective spaces

Generic Projection Methods in Castelnuovo Regularity of Projective Varieties

Castelnuovo-Mumford regularity is an important invariant of projective algebraic varieties. A well known conjecture due to Eisenbud and Goto gives a bound for regularity in terms of the codimension and degree,i.e., Castelnuovo-Mumford regularity of a given variety X is less than or equal to deg(X)-codimension(X)+1. Generic projection methods proved to be effective for the study of regularity of smooth projevtive varieties of dimension at most four(cf.[BM},[K2],[L],[Pi] and [R1]) because there are nice vanishing theorems for cohomology of vector bundles (e.g. the Kodaira-Kawamata-Viehweg vanish…

Geometric collections and Castelnuovo-Mumford Regularity

The paper begins by overviewing the basic facts on geometric exceptional collections. Then, we derive, for any coherent sheaf $\cF$ on a smooth projective variety with a geometric collection, two spectral sequences: the first one abuts to $\cF$ and the second one to its cohomology. The main goal of the paper is to generalize Castelnuovo-Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on smooth projective varieties $X$ with a geometric collection $σ$. We define the notion of regularity of a coherent sheaf $\cF$ on $X$ with respect to $σ$. We show that the basic …

On the Regularity of Powers of Ideal Sheaves

We use the geometry of the secant variety to an embedded smooth curve to prove some vanishing and regularity theorems for powers of ideal sheaves.

Projective surfaces of maximal sectional regularity

We study projective surfaces $X \subset \mathbb{P}^r$ (with $r \geq 5$) of maximal sectional regularity and degree $d > r$, hence surfaces for which the Castelnuovo-Mumford regularity $\reg(C)$ of a general hyperplane section curve $C = X \cap \mathbb{P}^{r-1}$ takes the maximally possible value $d-r+3$. We show that each of these surfaces is either a cone over a curve $C \subset \mathbb{P}^{r-1}$ of maximal regularity or else a birational outer linear projection of a smooth rational surface scroll $\widetilde{X} \subset \mathbb{P}^{d+1}$. We prove that the Castelnuovo-Mumford regularity of th…

Regularity of the Moduli Space of Instanton Bundles $MI_{\Bbb P^3}(5)$

We prove that the moduli space of mathematical instanton bundles on ${\Bbb P}^3$ with $c_2=5$ is smooth.

Castelnuovo-Mumford Regularity and Computing the de Rham Cohomology of Smooth Projective Varieties

We describe a parallel polynomial time algorithm for computing the topological Betti numbers of a smooth complex projective variety $X$. It is the first single exponential time algorithm for computing the Betti numbers of a significant class of complex varieties of arbitrary dimension. Our main theoretical result is that the Castelnuovo-Mumford regularity of the sheaf of differential $p$-forms on $X$ is bounded by $p(em+1)D$, where $e$, $m$, and $D$ are the maximal codimension, dimension, and degree, respectively, of all irreducible components of $X$. It follows that, for a union $V$ of generi…

对数-Lipschitz 与 Hölder 正则性蕴含复解析集的光滑性

Log-Lipschitz and Hölder regularity imply smoothness for complex analytic sets

本文证明任意维数与余维数下复解析集的对数-Lipschitz 及 Hölder 正则性均蕴含局部光滑性，并给出全局刻画。