Some properties of top graded local cohomology modules

This first part of the paper describes the support of top graded local cohomology modules. As a corrolary one obtains a simple criteria for the vanishing of these modules and also the fact that they have finitely many minimal primes. The second part of this paper constructs examples of cohomological Hilbert functions which are not of polynomial type

An Avoidance Principle with an Application to the Asymptotic Behaviour of Graded Local Cohomology

We present an Avoidance Principle for certain graded rings. As an application we fill a gap in the proof of a result by Brodmann, Rohrer and Sazeedeh about the antipolynomiality of the Hilbert-Samuel multiplicity of the graded components of the local cohomology modules of a finitely generated module over a Noetherian homogeneous ring with two-dimensional local base ring.Comment: 6 pages; to appear in Journal of Pure and Applied Algebra; corrected typo

Boundedness of Cohomology

Let $d \in \N$ and let \D^d denote the class of all pairs $(R,M)$ in which $R = \bigoplus_{n \in \N_0} R_n$ is a Noetherian homogeneous ring with Artinian base ring $R_0$ and such that $M$ is a finitely generated graded $R$-module of dimension $\leq d$. The cohomology table of a pair (R,M) \in \D^d is defined as the family of non-negative integers $d_M:= (d^i_M(n))_{(i,n) \in \N \times \Z}$. We say that a subclass $\mathcal{C}$ of \D^d is of finite cohomology if the set \{d_M \mid (R,M) \in \C\} is finite. A set $\mathbb{S} \subseteq \{0,... ,d-1\}\times \Z$ is said to bound cohomology, if for each family $(h^\sigma)_{\sigma \in \mathbb{S}}$ of non-negative integers, the class \{(R,M) \in \D^d\mid d^i_M(n) \leq h^{(i,n)} {for all} (i,n) \in \mathbb{S}\} is of finite cohomology. Our main result says that this is the case if and only if $\mathbb{S}$ contains a quasi diagonal, that is a set of the form $\{(i,n_i)| i=0,..., d-1\}$ with integers $n_0> n_1 > ... > n_{d-1}$. We draw a number of conclusions of this boundedness criterion.Comment: 18 page

Arithmetic properties of projective varieties of almost minimal degree

We study the arithmetic properties of projective varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely 2. We notably show, that such a variety $X \subset {\mathbb P}^r$ is either arithmetically normal (and arithmetically Gorenstein) or a projection of a variety of minimal degree $\tilde {X} \subset {\mathbb P}^{r + 1}$ from an appropriate point $p \in {\mathbb P}^{r + 1} \setminus \tilde {X}$. We focus on the latter situation and study $X$ by means of the projection $\tilde {X} \to X$. If $X$ is not arithmetically Cohen-Macaulay, the homogeneous coordinate ring $B$ of the projecting variety $\tilde {X}$ is the endomorphism ring of the canonical module $K(A)$ of the homogeneous coordinate ring $A$ of $X.$ If $X$ is non-normal and is maximally Del Pezzo, that is arithmetically Cohen-Macaulay but not arithmetically normal $B$ is just the graded integral closure of $A.$ It turns out, that the geometry of the projection $\tilde {X} \to X$ is governed by the arithmetic depth of $X$ in any case. We study in particular the case in which the projecting variety $\tilde {X} \subset {\mathbb P}^{r + 1}$ is a cone (over a) rational normal scroll. In this case $X$ is contained in a variety of minimal degree $Y \subset {\mathbb P}^r$ such that \codim_Y(X) = 1. We use this to approximate the Betti numbers of $X$. In addition we present several examples to illustrate our results and we draw some of the links to Fujita's classification of polarized varieties of $\Delta$-genus 1.Comment: corrected, revised version. J. Algebraic Geom., to appea

On varieties of almost minimal degree I: Secant loci of rational normal scrolls

To complete the classification theory and the structure theory of varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely 2, a natural approach is to investigate simple projections of varieties of minimal degree. Let $\tilde X \subset {\mathbb P}^{r+1}_K$ be a variety of minimal degree and of codimension at least 2, and consider $X_p = \pi_p (\tilde X) \subset {\mathbb P}^r_K$ where $p \in {\mathbb P}^{r+1}_K \backslash \tilde X$. By \cite{B-Sche}, it turns out that the cohomological and local properties of $X_p$ are governed by the secant locus $\Sigma_p (\tilde X)$ of $\tilde X$ with respect to $p$. Along these lines, the present paper is devoted to give a geometric description of the secant stratification of $\tilde X$, that is of the decomposition of ${\mathbb P}^{r+1}_K$ via the types of secant loci. We show that there are exactly six possibilities for the secant locus $\Sigma_p (\tilde X)$, and we precisely describe each stratum of the secant stratification of $\tilde X$, each of which turns out to be a quasi-projective variety. As an application, we obtain the classification of all non-normal Del Pezzo varieties by providing a complete list of pairs $(\tilde X, p)$ where $\tilde X \subset {\mathbb P}^{r+1}_K$ is a variety of minimal degree, $p$ is a closed point in $\mathbb P^{r+1}_K \setminus \tilde X$ and $X_p \subset {\mathbb P}^r _K$ is a Del Pezzo variety.Comment: 20 page

Castelnuovo-Mumford regularity of deficiency modules

Let $d \in \N$ and let $M$ be a finitely generated graded module of dimension $\leq d$ over a Noetherian homogeneous ring $R$ with local Artinian base ring $R_0$. Let \beg(M), \gendeg(M) and \reg(M) respectively denote the beginning, the generating degree and the Castelnuovo-Mumford regularity of $M$. If $i \in \N_0$ and $n \in Z$, let $d^i_M(n)$ denote the $R_0$-length of the $n$-th graded component of the $i$-th $R_+$-transform module $D^i_{R_+}(M)$ of $M$ and let $K^i(M)$ denote the $i$-th deficiency module of $M$. Our main result says, that \reg(K^i(M)) is bounded in terms of \beg(M) and the "diagonal values" $d^j_M(-j)$ with $j = 0,..., d-1$. As an application of this we get a number of further bounding results for \reg(K^i(M)).Comment: 25 pages, the previous version divided in two part

An example of an infinite set of associated primes of a local cohomology module

Let $(R,m)$ be a local Noetherian ring, let $I\subset R$ be any ideal and let $M$ be a finitely generated $R$-module. In 1990 Craig Huneke conjectured that the local cohomology modules $H^i_I(M)$ have finitely many associated primes for all $i$. In this paper I settle this conjecture by constructing a local cohomology module of a local $k$-algebra with an infinite set of associated primes, and I do this for any field $k$