1,594 research outputs found
Invariant hypersurfaces for derivations in positive characteristic
Let be an integral -algebra of finite type over an algebraically
closed field of characteristic . Given a collection of
-derivations on , that we interpret as algebraic vector fields on
, we study the group spanned by the hypersurfaces of
invariant for modulo the rational first integrals of .
We prove that this group is always a finite -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 -algebra between and
, we show that the kernel of the pull-back morphism is a finite -vector space. In particular, if is a
UFD, then the Picard group of is finite.Comment: 16 page
Cohomology of regular differential forms for affine curves
Let be a complex affine reduced curve, and denote by 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
that measures the complexity of the singularity of at the point
. Then, if denotes the first singular homology group of with
complex coefficients, we establish the following formula: Second we consider a family of curves given
by the fibres of a dominant morphism , where is an
irreducible complex affine surface. We analyze the behaviour of the function
. 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
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