Invariant hypersurfaces for derivations in positive characteristic

Let $A$ be an integral $k$-algebra of finite type over an algebraically closed field $k$ of characteristic $p>0$. Given a collection ${\cal{D}}$ of $k$-derivations on $A$, that we interpret as algebraic vector fields on $X=Spec(A)$, we study the group spanned by the hypersurfaces $V(f)$ of $X$ invariant for ${\cal{D}}$ modulo the rational first integrals of ${\cal{D}}$. We prove that this group is always a finite $\mathbb{Z}/p$-vector space, and we give an estimate for its dimension. This is to be related to the results of Jouanolou and others on the number of hypersurfaces invariant for a foliation of codimension 1. As an application, given a $k$-algebra $B$ between $A^p$ and $A$, we show that the kernel of the pull-back morphism $Pic(B)\rightarrow Pic(A)$ is a finite $\mathbb{Z}/p$-vector space. In particular, if $A$ is a UFD, then the Picard group of $B$ is finite.Comment: 16 page

Cohomology of regular differential forms for affine curves

Let $C$ be a complex affine reduced curve, and denote by $H^1(C)$ its first truncated cohomology group, i.e. the quotient of all regular differential 1-forms by exact 1-forms. First we introduce a nonnegative invariant $\mu'(C,x)$ that measures the complexity of the singularity of $C$ at the point $x$. Then, if $H_1(C)$ denotes the first singular homology group of $C$ with complex coefficients, we establish the following formula: $$dim H^1(C)=dim H_1(C) + \sum_{x\in C} \mu'(C,x)$$ Second we consider a family of curves given by the fibres of a dominant morphism $f:X\to \mathbb{C}$, where $X$ is an irreducible complex affine surface. We analyze the behaviour of the function $y\mapsto dim H^1(f^{-1}(y))$. More precisely, we show that it is constant on a Zariski open set, and that it is lower semi-continuous in general.Comment: 16 page

uFLIP: Understanding Flash IO Patterns

Does the advent of flash devices constitute a radical change for secondary storage? How should database systems adapt to this new form of secondary storage? Before we can answer these questions, we need to fully understand the performance characteristics of flash devices. More specifically, we want to establish what kind of IOs should be favored (or avoided) when designing algorithms and architectures for flash-based systems. In this paper, we focus on flash IO patterns, that capture relevant distribution of IOs in time and space, and our goal is to quantify their performance. We define uFLIP, a benchmark for measuring the response time of flash IO patterns. We also present a benchmarking methodology which takes into account the particular characteristics of flash devices. Finally, we present the results obtained by measuring eleven flash devices, and derive a set of design hints that should drive the development of flash-based systems on current devices.Comment: CIDR 200

On algebraic automorphisms and their rational invariants

Let X be an affine irreducible variety over an algebraically closed field k of characteristic zero. Given an automorphism Φ, we denote by k(X)Φ its field of invariants, i.e., the set of rational functions f on X such that f º Φ = f. Let n(Φ) be the transcendence degree of k(X)Φ over k. In this paper we study the class of automorphisms Φ of X for which n(Φ) = dim X - 1. More precisely, we show that under some conditions on X, every such automorphism is of the form Φ = ϕg, where ϕ is an algebraic action of a linear algebraic group G of dimension 1 on X, and where g belongs to G. As an application, we determine the conjugacy classes of automorphisms of the plane for which n(Φ) =

Minimal invariant varieties and first integrals for algebraic foliations

Abstract.: Let X be an irreducible algebraic variety over ℂ, endowed with an algebraic foliation {\user1{\mathcal{F}}} . In this paper, we introduce the notion of minimal invariant variety V( {\user1{\mathcal{F}}} , Y) with respect to ( {\user1{\mathcal{F}}} , Y), where Y is a subvariety of X. If Y = {x} is a smooth point where the foliation is regular, its minimal invariant variety is simply the Zariski closure of the leaf passing through x. First we prove that for very generic x, the varieties V( {\user1{\mathcal{F}}} , x) have the same dimension p. Second we generalize a result due to X. Gomez- Mont (see [G-M]). More precisely, we prove the existence of a dominant rational map F : X → Z, where Z has dimension (n − p), such that for very generic x, the Zariski closure of F−1(F(x)) is one and only one minimal invariant variety of a point. We end up with an example illustrating both result